OdedSchramm

OdedSchramm died September 1, 2008 (see Oded Schramm for further information). His talk page is preserved here.

Circle packing theorem

Hello. I've just added Circle packing theorem to the list of circle topics. If you know of other articles that should be listed there and are not, could you add those or point them out to me? Thanks. Michael Hardy (talk) 18:05, 30 March 2008 (UTC)

OK. I will. Oded (talk) 18:31, 6 April 2008 (UTC)

Topology Expert

Dear Oded,

I feel that there is some misunderstanding. I am not a "hoax" (what do you mean by this?), and my intentions are good. I am not the type of person who likes to boast, and I would rather people didn't know that I was a professor. That is why I edited your post. I am sorry that I left your signature (I didn't know that I did this). I earlier claimed that I was a professor because I felt people were not believeing my mathematical claims. However, after creating a few articles, I believe that I will gain more respect. Also, I keep on getting messages regarding me being a professor so that is also why I edited your post. I hope you understand. Secondly, my intentions are good. I have recently created three articles:

• Locally finite collection

• Locally connected space

• Linear continuum

All of these articles were either stubs or didn't have a page existent. I am also planning to do much more work on Wikipedia. Please see what I wrote on my talk regarding "supercompact spaces". I have also decided not to delete the article on supercompactness. Earlier, I wasn't familiar with Wikipedia and thought that only articles on major topics should be included. After seeing the page on "Supercompact spaces", I decided that I will also create pages on "minor"(though important!) topics and therefore created a page on linear continua. From now on I promise that I will be more understanding and not claim ownership of pages. I hope you understand me and that we will continue to collaborate in a friendly manner.

Topology Expert (talk) 01:25, 12 May 2008 (UTC)

Dear Topology expert: I am sorry that I called you a hoax. What I meant was that I thought that perhaps you are just trying to make fun of us. I can see now that your intentions are good and that you are doing good work. It is a bit hard to communicate with you. I am not sure if you are really reading what other editors have written to you. Finally, I see you no longer insist on including exercises in articles (we tried to explain this many times).

• Another point of style that is worth mentioning is that it is customary in wikipedia not to link every occurance of a technical term (such as open), but only the first one (or perhaps the first one in every section, in a long article). The idea is to link only if there is a good chance that the reader does not know the term.
• Also, when you state a theorem, or begin a proof, it makes sense to make them bold, so that they stand out. I.e. Theorem. Proof.

I hope that we can continue to communicate. Please let me know if you have read this. If I want to communicate with you - will you read what I write on your talk page? Oded (talk) 02:31, 12 May 2008 (UTC)

Dear Oded,

Thankyou for your response. I will definitely try to improve my article in terms of what you said. I was also wondering whether I could put my exercises some place else (if possible). I saw the wikibook on topology, and some things are incorrect. For example see the exercises in the section on local connectedness on Wikibook. I think that the Wikibooks could be improved in terms of exercises. Is it alright to put exercises there? Thanks for your help once again. Also, could you please write on my talk page next time? It is much easier for me to respond there.

Topology Expert (talk) 05:44, 12 May 2008 (UTC)

Dear Oded,

Thankyou for fixing up my article. I am not actually familiar with how to write mathematical symbols so I will have to learn. Also, the definition you gave for the deleted comb space is basically the same as mine, written in a different way. I noticed that you seem to be experienced in measure theory so perhaps you will know whether more pages are required on the subject. I don't know how much detail is necessary, but perhaps we could merge some related pages together. I think that the definition of an outer measure and the definition of the lebesgue measure fit together nicely so perhaps we could add a little bit about how the outer measure relates to the lebesgue measure? I am not so sure about this since I am not very familiar with Wikipedia. Could you please give me your opinion on this?

I also wanted to ask whether I could change the name of the article on 'locally connected spaces'. When I first created this article I thought that this concept has a connection with the concept of components of a topological space so I decided to add that in. I don't think there is a page written on 'components' anyway so maybe we could change the name of the article to 'Components and locally connected spaces' which is a much more appropriate name. Could you please tell me how to do this?

I also noticed that there is not much written on the concept of the uniform norm. They have only used this in the context of function spaces but perhaps the more general reader would also prefer a view relating to the product topology. I am happy to write a page on that if necessary. I was also thinking that (as you mentioned) pages on concepts such as 'mesacompactness' and 'orthocompactness' could be improved. They are indeed important in mathematics. I am looking to improve pages on more elementary concepts which do turn out to be important in other branches of mathematics.

Thankyou for your help once again.

Topology Expert (talk) 10:47, 13 May 2008 (UTC)

Dear Oded,

Sorry for the late response. I meant (when I said that I was going to add a page on the uniform topology), was that there are pages on how the uniform topology fits in the context of functions spaces, but there is no page relating to the uniform topology on R^{ωSuperscript text}. Some readers (particularly students), may prefer to read about the properties of the uniform topology on R^{ωSuperscript text}. Of course, I am not saying that we should delete the original pages, but maybe we can add a page on this. I am not so sure whether Wikipedia is a learning tool (i.e, information should be written with examples so that people can learn), so maybe we shouldn't add an extra page relating to the uniform topology on R^{ωSuperscript text}. But in my opinion, some people may want to also read how the uniform topology can be imposed on R^{ωSuperscript text} and some of its properties. Of course, the uniform topology is most used in functional analysis. Could you please give me your opinion on this?

Also, I recently added some information regarding the relevance of the 'induced homomorphism' in algebraic topology (about two pages on a word file). But, David Eppstein deleted what I wrote. Therefore, I asked him why he did this but perhaps you may know what was wrong with what I wrote (since you are familiar with Wikipedia). Thankyou for your help.

Topology Expert (talk) 08:47, 17 May 2008 (UTC)

Dear Oded,

I wasn't stating my intentions clearly; sorry for that. I really meant 'uniform metric' instead of 'uniform norm' so I did make a mistake. I (having learnt topology first from Munkres's text book), believe that perhaps we should include a new page on the uniform topology in relation to R^ω and other spaces (namely products of metric spaces). I feel opposed to my original intention in which I thought that beginners may find it beneficial to read a more elementary view of the uniform topology. Perhaps a functional analysis point of view may be a bit irrelevant for someone learning point-set topology. Now I see that Wikipedia is not for learning so perhaps it is not a good idea to include such a page. On the other hand, you mentioned earlier that Wikipedia is about putting maximum detail so I guess it is not such a bad idea. Could I please have your opinion on this? Thankyou for all your help.

Topology Expert (talk) 10:00, 19 May 2008 (UTC)

Uniform Topology

Dear Oded,

I wasn't able to respond to your previous message because of other committments. Sorry about that. I noticed that you removed the section on the local finiteness of topological spaces in the article 'locally finite collection'. I completely agree that it should be removed. But you wrote that the sentence, 'every locally finite space is finite' is false. According to the definition that was written under the section, this should be true. For example, the power set of an infinite topological space, is an example of a collection which is not locally finite (because it isn't point-finite). So basically, I was wondering why you wrote that the last sentence was false. I think that that section should be deleted anyway.

Also, the uniform topology does appear to have a name in textbooks. In the book by Munkres, it is referred to as the uniform topology and the metric that induces this topology is referred to as the uniform metric. As most people study the book by Munkres, it seems appropriate to title the article as 'uniform topology'. I was thinking of perhaps creating this article under this name. Perhaps you would know whether this should be done.

I also recently thought of some additional concepts that could be included in the article on 'Locally connected space'. Namely, the concept of a weakly locally connected space and the notion of quasicomponents. I feel that by adding this, I am including a bit too much information in this article. Maybe it would be better to split this article up. On a word file it takes up 7 pages which I don't think is appropriate for an article on this concept. On the other hand, there are many more things that can be added under this title and I am ready to add them. Could you please give me our opinion on this?

Thankyou for your help.

Topology Expert (talk) 08:00, 3 June 2008 (UTC)

The space Z (integers) is locally finite but infinite.

With regards to your other questions, I'll reply soon. --Oded (talk) 18:32, 3 June 2008 (UTC)

I don't not have a definite opinion as to whether we should have an article on the uniform topology, beyond the way it is currently covered. If all there is to say about the subject is what I know, then the answer would be no. Oded (talk) 05:12, 4 June 2008 (UTC)

Dear Oded,

I know this is a trivial matter, but according to the definitions I have seen (a space is locally finite if every collection of subsets of the space is locally finite), the integers shouldn't be locally finie. This is because the collection of all open subsets of the set of integers is not locally finite (because this collection isn't point finite). If you are correct, then the definition I have read is wrong. On the other hand, many reliable sources give the same definition as what I know. So I was just wondering how the definition you know is worded. Most probably, the definition I have got is wrong.

I thought about how the article on supercompacntess could be improved and I have some ideas. I was thinking of including some more spaces that are supercompact in the article. However, I am unsure whether I should do this. From what I have seen in mathematics articles, it is alright to include certain types of topological spaces that are supercompact. I am going ahead with this. If what I am doing is inappropriate for the article, could you please let me know?

Topology Expert (talk) 07:55, 4 June 2008 (UTC)

You are right about locally finiteness. I was a bit confused.

Regarding supercompactness: I think you initially wanted to delete the article, as you thought it was not important enough. Since I don't know much about supercompatness, I don't feel that I can give advice as to what to add to the article. Likewise, I would think that unless you have now become a supercompactness enthusiast and feel that you have a good idea of what are some of the important facts known (published) about supercompactness, that you should probably leave the possible expansion of the article to those who have the subject closer to their heart.

Oded (talk) 09:57, 4 June 2008 (UTC)

Dear Oded,

About local finiteness of a topological space, I believe that the following is a more appropriate defintion (in my opinion):

A topological space, X, is said to be locally finite, if every collection of disjoint subsets of X is locally finite.

The requirement that the sets be disjoint solves the problem and makes the statement, 'a space is locally finite iff it is finite' false. In this case, the claim that the integers is locally finite is true. However, most mathematics sources give a different definition. Truthfully speaking, the integers should be locally finite according to definition but isn't. In this definition it is. What is your opinion on this definition?

I happened to notice that there is no article on the broom space. I think that a definition of the broom space is worthwile in the context of weakly locally connected spaces (or possibly in other contexts) but I am not sure. Do you think that an article should be created on the broom space?

Thanks for your help.

Topology Expert (talk) 06:42, 8 June 2008 (UTC)

I don't think that there is any need to define a locally finite space. The first variant on the definition is just finite spaces, while the second variant is the same as being a discrete space (space with the discrete topology).

I don't know the broom space.

Oded (talk) 16:24, 8 June 2008 (UTC)

Countable set

Dear Oded,

Someone named JRSpriggs called what I added to the page on countable set 'incompetent and irrelevant'. This is what I added to the page (when I typed this on Wikipedia I did use TeX):

---

Topological proof of the uncountability of the real numbers

Theorem

Let X be a compact Haudorff space that has satisfies the property that no one point sets are open. If X has more than one point, then X is uncountable.

Before proving this, we give some examples:

1. We cannot eliminate the Hausdorff condition; a countable set with the indiscrete topology is compact, has more than one point, and satisfies the property that one point sets are open, but is not uncountable.

2. We cannot eliminate the compactness condition as the set of all rational numbers shows.

3. We cannot eliminate the condition that one point sets cannot be open as a finite space given the discrete topology shows.

Proof of theorem:

Let X be a compact Haudorff space. We will show that if U is a nonempty, open subset of X and if x is a point of X, then there is a neighbourhood V contained in U whose closure doesn’t contain x (x may or may not be in U). First of all, choose y in U different from x (if x is in U, then there must exist such a y for otherwise U would be an open one point set; if x isn’t in U, this is possible since U is nonempty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively. Then (W intersection U) will be a neighbourhood of y contained in U whose closure doesn’t contain x as desired.

Now suppose f is a bijective function from Z (the positive integers) to X. Denote the points of the image of Z under f as {x₁, x₂, ……}. Let X be the first open set and choose a neighbourhood U₁ contained in X whose closure doesn’t contain x₁. Secondly, choose a neighbourhood U₂ contained in U₁ whose closure doesn’t contain x₂. Continue this process whereby choosing a neighbourhood U_n+1 contained in U_n whose closure doesn’t contain x_n+1. Note that the collection {U_i} for i in the positive integers satisfies the f.i.c and hence the intersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection. No x_i can belong to this intersection because x_i doesn’t belong to the closure of U_i. This means that x is not equal to x_i for all i and f is not surjective; a contradiction. Therefore, X is uncountable.

Corallary

Every closed interval [a,b] (a<b) is uncountable. Therefore, the set of real numbers is uncountable.

---

I am pretty sure that this is correct an there is nothing wrong with the proof. Why would he call this section 'incompetent'? Also, this is in the article on countable set so it is relevant. Could you please check the edit history and see whether this user has presented a valid reason? In my opinion, he has given an invalid reason.

Thanks

Topology Expert (talk) 08:46, 16 June 2008 (UTC)

I think I mostly agree with JRSpriggs' edit, though his words were perhaps too harsh. The article is about countable sets. But the theorem you put in is about an uncountable set. The theorem would fit in better in the article continuum. Certainly, the discussion of the continuum hypothesis and GCH does not belong in the countable set article. With regards to reasons why JRSpriggs might characterize this as incompetent, perhaps some of the reasons are: 1. the mistake in stating the GCH, 2. the fact that there are already articles on the continuum hypothesis and this is just duplication of material in an unfitting location, 3. the set of positive integers is denoted there by $Z$, rather than the customary $\mathbb {N}$, 4. there is a link to a non-article f.i.c, and there certainly should not be such an article and there is no reason to believe the reader would know what this means unless they can write the article themselves, 5. the WP style is not sufficiently respected. Oded (talk) 15:10, 16 June 2008 (UTC)

Dear Oded,

I did make a few mistakes such as giving an incorrect statement of the GCH, which I should have been careful about. But since the integers (Z) have the same cardinality as the natural numbers (N), it shouldn't really matter what symbol I use (I used the symbol in the context of countability; I could have chosen any old countable set for the proof). Perhaps what I wrote on the continuum hypothesis should be deleted but I think that the other section should be kept. I agree with the conditions you put forth, but those things can be easily fixed (just change f.i.c to Finite intersection condition). For the fifth condition, could you please tell me what was wrong with my style of writing? I feel that the section I added should be bought back but I want to have your opinion just in case.

Thanks

Topology Expert (talk) 06:53, 17 June 2008 (UTC)

The previous post was written by me (I wrote it without logging in); sorry about that. Please ignore the part about changing f.i.c to Finite intersection condition. Is there an article on the finite intersection condition?

Topology Expert (talk) 06:53, 17 June 2008 (UTC)

Dear Topo,

• There is the article finite intersection property.
• Getting notations that are consistent with the standards is important.
• I guess if you re-read carefully your edits, you would be likely to catch such mishaps.
• The topological proof of the uncountability of the reals does not belong in the countable set article. If anywhere, I think it would belong in the continuum article. But I think that actually for the continuum article a proof based on the linear order of the reals would be nicer.
• With regards to style, you asked for more specify criticism. First, I think that enumerating points as you have would not be appropriate in this case. Second, the repeated use of we is certainly discouraged. Third, given that the article is not in topology, you should have linked notions such as open and neighborhood. Which brings us back to the fact that the argument does not belong there.

Oded (talk) 14:49, 17 June 2008 (UTC)

Dear Oded,

I understand what you are trying to say, but instead of completely removing what I wrote, wouldn't it be better to edit the mistakes? If the problem is related to the fact that my edit shouldn't belong there, then where can it belong to? I am going to add it to the section on compact spaces. Could you please tell me if I shouldn't or if there is a better place to add it to.

Thanks

Topology Expert (talk) 09:45, 20 June 2008 (UTC)

Perhaps I should add it to the continuum (mathematics) articles as you suggested. I will just read both articles and see which is more appropriate. However, do you think it is an option to create a new article on this proof and other proofs too?

Topology Expert (talk) 09:49, 20 June 2008 (UTC)

It would not be a good idea to add this in either of these places. It is not one of the simplest or most direct proofs. This proof is very similar to the proof of the Baire category theorem. It would be good if you would adapt this to prove one of the two versions of the Baire theorem that are stated there (or even a weaker statement, simplicity would be more important than generality). Then in the article continuum, you could mention that the Baire category theorem easily implies that the real numbers are not countable.

In general, it is more fruitful to think what would be best to improve an article or how to create the best possible article, rather than asking "in which article would this and that fit best". Oded (talk) 05:49, 22 June 2008 (UTC)

Hausdorff Dimension

Dear Oded,

I made some mistakes when I edited a couple of articles but I will fix them up.

Topology Expert (talk) 08:37, 23 June 2008 (UTC)

Deletion of the Tube Lemma

Dear Oded,

Please see the page on the Tube lemma. I have nominated it for deletion for several reasons which are given on the discussion page. The main reason is probably the fact that the article doesn't even hint that there is a relation between the tube lemma and compactness but there are other reasons too.

Thanks

Topology Expert (talk) 11:11, 23 June 2008 (UTC)

Almost Surely

Dear Oded,

Could you please justify your claim that a nowhere dense subset of the plane can have positive measure?

Thanks

Topology Expert (talk) 11:36, 23 June 2008 (UTC)

See Nowhere dense#Nowhere dense sets with positive measure. I can also find sources outside of Wikipedia; it is well known that there are simple paths in the plane with positive measure and they are of course nowhere dense. Why did you think that nowhere dense sets have zero measure? Oded (talk) 16:01, 23 June 2008 (UTC)

Problem

Dear Oded,

I am not able to type on your talk page; whenever I try, everything else gets removed. Could you please see my talk page instead (the last section)?

Thanks

Topology Expert (talk) 06:28, 24 June 2008 (UTC)

Wikipedia

Dear Oded,

First of all, I was in a bit of a hurry when I wrote that a nowhere dense set has zero measure and I didn't write what I really meant; of course that is no excuse for what I wrote. What I really meant was that if a set has zero Lebesgue measure, then it must be nowhere dense.

Second of all, for the article on Hausdorff dimension, I initially wrote that a countable discrete space has Hausdorff dimension 0. You then said that this was wrong so I changed it (because I believed you) to something which was really wrong. If you don't believe me see the history for that article.

I also think it is quite unreasonable of you to say that I make mistakes often. It is not like you haven't made mistakes before. I find it quite insulting that you say I 'make up' mathematics and put it on Wikipedia.

Topology Expert (talk) 09:36, 25 June 2008 (UTC)

Not so. In Hausdorff dimension you initially wrote that every discrete space has Hausdorff dimension 0. Did you find this in a textbook? In an article? I assume not. Therefore, I assume that you came to the conclusion that this is a correct statement and then you put it in. Then you wrote that a countable discrete space has Hausdorff dimension 1. Again, this must be your own "discovery". It is perfectly ok to make mistakes every once in a while. But I think that your rate of making mistakes indicates that you should be much more careful and very closely follow books and articles. Contrary to what you wrote above, it is also not true that a set of zero Lebesgue measure is nowhere dense. That would also have no relevance to the article you were editing at the time.

You may ask why I removed your latest entry from Hausdorff dimension. The first example says that countable sets have zero Hausdorff dimension, and there is no reason to repeat an obscure particular case. Also, this fact that the Hausdorff dimension of countable sets is zero is rather marginal in this article, and it would not make any sense to talk so much about it. Moreover, the more general and useful thing to mention is that a countable union of sets has Hausdorff dimension that is equal to the supremum of the Hausdorff dimensions of the sets. Oded (talk) 15:42, 25 June 2008 (UTC)

First of all consider the statement:

Every elementary set is finite m-measurable (m is the Lebesgue measure)

One does not need to give a reference of this fact in order to add it to Wikipedia. The case for which a countable discrete space has Hausdorff dimension 0 is analagous. If I am adding a theorem which is either famous or published then I will give a reference.

Also, I am quite certain that a set of Lebesgue measure 0 is nowhere dense. If you accept that the Lebesgue measure is an additive set function then it follows that if A is Lebsgue measurable, and B is Lebesgue measurable and A is a subset of B, then m(A) <= m(B). Then noting that every basis element in the plane (or R^n for any positive integer n) has positive measure, it follows that if X is a set of measure 0, X must be nowhere dense (if X did contain a basis element, it would have to have greater measure than this basis element which means it would have positive measure).

I also can't see where I am making mistakes (apart from writing that a nowhere dense set has Lebesgue measure 0 and forgetting to add that a countable discrete space has Hausdorff measure 0 and not an arbitrary discrete space). I don't really care if you remove my entries since I am not writing them for my benefit.

Topology Expert (talk) 06:21, 26 June 2008 (UTC)

Regarding our disagreement as to whether or not a set of Lebesgue measure 0 i nowhere dense, I think that we are following different definitions. The definition I follow is:

A subset Y of a topological space X is nowhere dense if and only if the interior of Y is empty.

The definition you probably follow is:

A subset Y of a topological space X is nowhere dense if and only if the interior of the closure of Y is empty.

Note that the definition I follow is from Munkres' book (when I studied it) so I think that the definition I follow is correct. But from several other sources it seems that your definition is correct. Therefore, I think it is fair to say that neither of us was wrong.

Topology Expert (talk) 06:24, 26 June 2008 (UTC)

Today I had to undo 3 of your edits. Each of them were incorrect as stated. In paracompact space, you neglected to restrict to infinite sets. (Moreover, the particular point topology is not interesting enough to mention there.) In Category (mathematics) you said that a fibre bundle is a category, which is false or meaningless. In compact space you said that in a linearly ordered set with the least upper bound property a closed interval is compact. A counterexample is $(-\infty ,0]$. I did not check your edits to the tube lemma.

Perhaps Munkres says that "a closed set is nowhere dense if it has empty interior". I cannot imagine that according to Munkres the rationals are nowhere dense in the reals. Can you quote Munkres' precise definition? In any case, when editing WP you should generally try to follow the definitions given in WP. Oded (talk) 15:41, 26 June 2008 (UTC)

You continually make mistakes. How is (-infinity, 0] a counterexample? Infinity is not an element of the set; it is the infimum of the set (if you want a reference to this, I can give you one). I want you to precisely tell me which closed interval is not compact in (-infinity,0]. You seem to only target my edits and revert them (how come you are able to pinpoint which articles I have edited and change them); could you please stop doing this? Instead, go and revert edits which are really wrong.

Topology Expert (talk) 06:18, 27 June 2008 (UTC)

That's what I am trying to do. The interval $(-\infty ,0]$ is closed in $\mathbb {R}$, but not compact. Oded (talk) 06:32, 27 June 2008 (UTC)

Dear Oded,

Apologies for what I said earlier. If you read what I wrote carefully, then you would have seen the following statement:

Let X be a simply ordered set having the least upper bound property. If X carries the order topology, then every closed interval in X is compact.

Notice that it is written that every closed interval in X is compact. The example you have given $(-\infty ,0]$ is not a closed interval in $\mathbb {R}$. I hope you understand what I am trying to say. Please understand that everyone makes mistakes and although I have made a few (compared to how much I have written), it doesn't mean that everything I write is wrong. Anyway, I hope that we can sort out this 'problem' and continue as before.

Topology Expert (talk) 11:02, 27 June 2008 (UTC)

Note that even $\mathbb {R}$ itself is a closed interval in $\mathbb {R}$, since it is an interval and a closed set (see interval (mathematics)). It is not compact. On the discussion page of tube lemma I indicated some problems with that article. Please have a look and fix them. Also, your changes to the lede in the article limit point compact do not provide an illuminating example, since according to the article every metrizable limit point compact space is compact. Oded (talk) 15:43, 27 June 2008 (UTC)

I reverted my deletion of the examples, since my example was flawed. Back in the 20th century when I learned topology, I think we called closed intervals only those of the form [a,b]. Now "closed interval" meaning "an interval that is closed" sounds quite a bit saner, and agrees with the wiki article. If T.E. only mean that [a,b]'s were closed, then maybe he can be more specific. However, I definitely think T.E. should give an {{inline}} citation to a source, otherwise the content should probably still be deleted as unsourced additions.

Out of curiosity though, is there a simply ordered space with the least upper bound property containing two elements a,b such that the interval [a,b] is not compact? JackSchmidt (talk) 18:03, 27 June 2008 (UTC)

I convinced myself that in a linearly ordered space with least upper bounds intervals of the form $[a,b]$ are compact with respect to the order topology. Topology expert was confused about the meaning of the word closed in closed interval. On the one hand, he thought it meant an interval of the form $[a,b]$, while on the other hand he wrote "with respect to the order topology", which means that the interval is topologically closed. Either way, ambiguously correct/false statements are worse than no statements at all (almost always). Oded (talk) 00:55, 28 June 2008 (UTC)

Articles

Dear Oded,

I have a few things to point out:

1. I found a reference to the statement I wrote earlier regarding compactness in the order topology; see the book by Munkres. Since I have given a reference you will probably believe me know.

2. I didn't write the proof of the tube lemma; I wrote everything else so I can't really do much about that. I can give another proof if necessary.

3. I also don't see what is so hard to understand about example 1 (I think Jack fixed that up and made it more clear)

4. If you read the proof of the Tychonoff theorem, it uses two key concepts; the maximum principle and the fact that the finite intersection condition is equivalent to compactness.

5. I am not 'confused' about what a closed interval is. I can name numerous books which define the closed interval as I have.

6. My goal is to improve Wikipedia and not to write incorrect statements. You seem to critizise everything I write without giving a valid reason. So far, your reasons have not been appropriate to the context and I think that you should see what good I am doing instead of seeing what bad I am doing.

Topology Expert (talk) 03:09, 28 June 2008 (UTC)

Perhaps you can find a book which defines closed interval to be an interval of the form $[a,b]$. But regardless, this is not the way it is defined in WP, and there is no reason why the reader would interpret it this way. If you write something like

"Let X be a simply ordered set having the least upper bound property. Then Every closed interval in X is compact"

then you should make sure that all the terms you are using are either clear from the context (i.e., they have already been discussed or there is no reason a reader would be reading the article unless they know this) or that you are giving sufficient pointers to see what they mean, and that the way you are using them is consistent with the links you give (and hopefully also consistent with the standard usage).

Just because some statement is true is not reason enough to add it to an article. You should be thinking more about exposition. The articles are not lists of facts. They are meant to give an overview of the most important aspects of the concept covered.

Perhaps the proof in the article of the Tube lemma was more understandable before you made your edits. I have not checked this, so I don't know.

You are right that I usually point out the bad rather than the good. Perhaps this is unfortunate. The bad needs fixing, and the good is good.

I still suggest that you try to be more careful in your edits and check that your statements are correct and unambiguous. Also, please try to think more about good exposition, relevance, and order.

Oded (talk) 03:48, 28 June 2008 (UTC)

PS. I did read a proof of Tychonoff's theorem, but there was no maxmimum principle in it. If you look up maximum principle, you will find a concept which I'm pretty sure has nothing to do with Tychonoff's theorem. Do you mean Zorn's lemma?

Dear Oded,

It is a bit difficult for me to properly structure an article that I haven't created mainly because if I edit some other article, I have to follow the style of that article and not my own. This is one of the reasons why I prefer to create articles instead of editing old ones. However, the article on the tube lemma wasn't in a good state and I improved it in the hope that someone may build on that. Anyway, if the proof of the tube lemma doesn't fit in with what I wrote, I can give a different proof but I think for the time being it is best that I leave the article as it is.

Regarding good exposition, you might notice that articles I create generally have a better structure than what I write on other articles. I stick to the guideline that if an article has been created, I will add relevant facts, fix links or correct mistakes; I may even add a section. When I create an article, I have lots of freedom because I know what my intentions are and I am able to properly 'fit in' facts so that the article retains its good exposition. For the tube lemma, I thought that I will properly structure what I can and this is what I did.

I have also noticed that many of the definitions in Wikipedia do not agree with the definitions in textbooks. Anyway, I will follow the definitions given in Wikipedia from now on.

Thanks

Topology Expert (talk) 06:18, 28 June 2008 (UTC)

Dear Oded,

I think that there are quite a few proofs of the Tychonoff theorem. However, the proof I know does use the maximum principle. I will let you know from which source I obtained the proof soon. However, I think that it was from a textbook.

Topology Expert (talk) 06:21, 28 June 2008 (UTC)

But what do you really mean by the "maximum principle"? Do you mean the principle that analytic and harmonic functions take the maximum value at the boundary? (This holds more generally to other solutions of PDE's). I doubt it.

Indeed, the proof of the Tube lemma did fit to the statement that was before. That is another thing that one must be careful about when editing - the rest of the article has to still make sense or be modified accordingly. Oded (talk) 06:24, 28 June 2008 (UTC)

Dear Oded,

I saw the article in Wikipedia on the maximum principle. I am not actually referring to this article (it certainly has nothing to do with the Tychonoff theorem!); I am referring to the maximum principle in set theory.

Topology Expert (talk) 06:26, 28 June 2008 (UTC)

I think that in Wikipedia, it is called the Hausdorff maximality theorem. Also, the proof of the Tychonoff theorem using the maximal principle can be found in Munkres' textbook on topology.

Topology Expert (talk) 06:31, 28 June 2008 (UTC)

What about this

Hi Oded!

Anyone can make a mistake, you know... If you think you know better and that there's no need to be friendly with other contributors just because you're right, you might be interested in looking for alternative venues [1], [2]. Happy editing! Expert in topology (talk) 11:27, 29 June 2008 (UTC)

Yes, it is sometimes frustrating having your work sometimes carelessly messed up by others. But I think that Topology expert was the only user/editor that I have ever had a serious disagreement with. There is also quite a bit of satisfaction working with others who are competent and knowledgeable. We each build on the strengths of others. Oded (talk) 20:56, 29 June 2008 (UTC)

On the contrary, looking over Oded's contributions, I think his handling of disputes has been exemplary. He has shown a great deal of patience. His willingness and ability to clearly explain his edits and comments is commendable. I think you are confusing being "unfriendly" with "explaining elementary errors patiently". Everybody makes mistakes, and I'm sure Oded understands that. If a mistake is pointed out to you, note it and move on. There is no need to carry around bruised pride or make snipes at people who correct you. If this kind of thing happens a lot, that may be a sign that more care is needed or perhaps restraint in editing those topics. By the way, why is your user name so similar to Topology Expert? I don't think you are the same person. --C S (talk) 00:07, 30 June 2008 (UTC)

[edit conflict] Your answers are welcome. Just please note that disagreements are our life here. In most cases you would get it right and often your explanations are very detailed. Certainly, that's OK. My suggestion would be just to try to apply a more editor-friendly approach and give up kind of personal approach and condescending style that accompanies some corrections (I could be more specific on request but find it not that helpful here). I believe that such approach is not welcome on the wiki, esp. when you implicitly use some academic credentials. Maybe some editorial experience makes it difficult. Well, if frustration prevents you from discussing in a friendly manner with other Wikipedians... you know. Anyway, happy editing.

BTW, I explained in details what's wrong with your recent reverts in Hausdorff measure. While there are some questions of style (that might be subjective and open to debate) I believe I removed some (objective) inaccuracies. Expert in topology (talk) 09:15, 30 June 2008 (UTC)

PS. The nick was just to indirectly support (the spirit of) Topology Expert. It looks that it worked (see my talk if you like). EiT.

Did you know that sock puppeting is not permitted? Oded (talk) 15:56, 30 June 2008 (UTC)

What makes you think that there are illegitimate sock puppets (see C S post)? Let's make a peace. In fact I'm hoping to abandon this nick today. Expert in topology (talk) 17:06, 30 June 2008 (UTC)

I did not think we were ever at war. I regret that you got upset about me saying that you make frequent mistakes. I never reverted any of your edits because of anger. (Sometimes, I was too lazy to sort out your bad modifications from your good ones, and therefore reverted the entire edit, as I thought that on the whole it was not helpful.) I am certainly willing to explain any of the reverts and edits that I have made. (Please list those that you want explained/justified). Oded (talk) 17:28, 30 June 2008 (UTC)

I'm quitting in peace, as planned before. A bit sad, because of your recent initiative, you know. The more that no war declaration has just been issued here. And we made a substantial progress with the Hausdorff measure article. There were really two different and unrelated "topologists". Sorry, if my nick made you upset. Happy editing, anyway. Expert in topology (talk) 21:44, 30 June 2008 (UTC)

Dear All,

I think that it is a bit unfair to say that Oded was being unfriedly but thanks for the support. Also, I think that the main disagreement between Oded and I is that we sometimes follow different definitions; I think that Oded agrees with this. Also, regarding the Tychonoff theorem, I think that Oded has to use common sense. I said that the Tychonoff theorem uses the maximum principle in the proof. Oded said that this was bogus. There are two maximum principles; one relating to complex analysis and one relating to set theory. I think that common sense should prevail in deciding which maximum principle relates to the Tychonoff theorem. I just want to know why you (Oded) thought that I was referring to the maximum principle in complex analysis. Could you please tell me?

I forgot about the Hausdorff maximal principle. In this case your statement was correct, but I expect it would not be useful for many readers if you do not supply a link to where the concept is explained. Oded (talk) 15:53, 30 June 2008 (UTC)

Also, I am happy to accept mistakes and I don't think that someone is unfriendly by saying that I make mistakes so please don't think that. It is just that Oded should wait to see what I have got to say regarding the mistake before reverting my edit. Oded should understand that I am also a mathematician.

Sometimes, the mistake is not really a mistake and it is because we follow different definitions as I mentioned earlier. Second of all, I don't make mistakes everyday and some things I write are correct but Oded reverts it because he follows a different definition. Could you (Oded) please check whether the 'mistake' may be due to us following different definitions and not because it really is a mistake? Anyway, I hope that you all agree, that Oded understands what I amd trying to say and that we will continue to communicate.

Topology Expert (talk) 07:38, 30 June 2008 (UTC)

Dear Oded,

I also don't understand why you say that the tube lemma is not related to the Tychonoff theorem. Don't you think that it is worthwile to mention that the tube lemma proves that the product of finitely many compact spaces is compact and that the Tychonoff theorem proves that the product of arbitrarily many compact spaces is compact? Also, I think that the reader will find it useful to note that the tube lemma doesn't generalize to infinite products.

Topology Expert (talk) 07:45, 30 June 2008 (UTC)

All I was saying was that there is no point to explain in tube lemma what methods of proof can be used to prove Tychnoff's theorem. It is relevant that Tychonoff's theorem generalizes some of the applications of the tube lemma and that the tube lemma cannot prove it, but starting to say in tube lemma what methods will prove Tychonoff is probably out of place. Oded (talk) 15:53, 30 June 2008 (UTC)

Dear Oded,

I understand what you are trying to say regarding the Tychonoff theorem. Also, I am not Expert in Topology, I don't know who he is nor do I know why he created an account. If you notice, my style of writing is completely different from his.

Thanks

Topology Expert (talk) 00:33, 1 July 2008 (UTC)

Sock Puppetry

Dear Oded,

Please understand the following:

What if someone starts an account by the name 'Oded1' and writes exactly like you and you know that you are not 'Oded1'? Then what if someone accuses you of creating a new account? What would you do? I am in the exactly same situation. I don't know who started an account 'expert in topology' and why he did it. In fact, I think that his purpose was to get me in trouble.

How could you think that I have done this? I never new that you could do such a thing to someone else. I want to expand Wikipedia and creating multiple accounts will get me nowhere.

Suppose I did do this. Then why would I choose expert in topology; isn't that stupid since the name is so similar to topology expert? You have got to believe that I am not expert in topology. I also think that this is totally unfair since you have got no concrete evidence. What if someone writes in exactly the same style as you; how can you defend yourself? What are you intentions anyway?

Topology Expert (talk) 06:34, 4 July 2008 (UTC)

Dear Oded,

I think that I did get a bit angry about the matter. Please understand that I am not 'expert in topology' and that I didn't even think that it was possible to create another account for this purpose. I originally thought that 'expert in topology' was supporting me so I thanked him. Then I realised that he is doing this to get me in trouble. I was a bit unhappy about some of your edits but I never would have imagined doing this. Also, do you think that creating another account will convince you (if at all there is some reason to convince you)? If you think that I am 'sock puppeting' then that is fine but please consider the possibility that someone else created an account for this purpose. Also, I hope that we continue to collaborate in a friendly manner and that we can ignore what has happened in the past.

Topology Expert (talk) 06:53, 4 July 2008 (UTC)

Dishonesty

I find it very hard to imagine a explanation in which T.E. has not been dishonest, especially in his early days, but I think he may still become a valuable contributor. As he becomes more invested in the project and more familiar with the people here, I think his earlier mistakes will either be silently forgotten or he will himself admit them, and move on. I am a little worried that the sock case is not a good place for him to do this though; here he is being accused, and by human nature must be defensive. I think for him to move beyond earlier mistakes, he'll have to feel part of the community and see he has a place here (rather than trying to defend his place).

If you are definitely convinced EiT and TE are the same person (or collaborating from the same place), it is probably best to ask for a checkuser to set our minds at ease (one way or the other). I believe sufficient justification for the CU would come from TEs persistent anon edits, sometimes only a minute before and after a logged in edit.

I try to stay out of the actual content dispute, since this is a combination of mathematical correctness, mathematical taste, and encyclopedic taste. In case it is not clear, I very much support you in matters of mathematical correctness, and I definitely want to thank you for keeping our articles from being sloppy collections of mathematical lies.

I think however that it is dangerous at wikipedia to try and go after the "truth", rather than "verifiability". If I really care whether or not something is true, I work it out on my own and then have an expert check my work. All I expect from wikipedia is that the lies it tells are the standard ones (for instance, I still have not deleted a subgroup lattice picture of the sporadic simple groups that is just wrong, because it appears in a reliable source. Instead I just marked the image discussion with the erratum). Basically, the arguments are going to be long and fruitless if you argue about truth, but if you argue about WP:V and WP:RS, all admins will be on your side, and the other editor will be forced to reread his sources, and hopefully correct his own mistake.

We only have a limited number of editors who are really familiar with topology, and we definitely appreciate your work. On a personal note, I wanted to thank you for being patient during the order topology question, as it made for several enjoyable hours of me relearning a bit of point set topology (I think the last I had looked at order topology was 1996... wow, apparently I didn't learn it very well then). If you are interested, I have a series of naive but hopefully interesting questions I was going to ask at WP:RD/MATH once the dispute has blown over. It occurred to me that the subspace topology on {0} union (1,2) union {3}, as a subspace of the real line, cannot be induced from an order topology. I wanted to give an algebraic proof using Aut() in various categories, but couldn't make it stick. I also wondered about various cardinalities of discrete sets; are only countable cardinalities induced from order topologies? I suspect there is some embedding theorem due to a Polish guy that would answer most of my questions. JackSchmidt (talk) 21:31, 4 July 2008 (UTC)

I'll try to reply to all your comments. I do appreciate you trying to make things work better. Perhaps the sock reporting was not the right thing. I'm not sure. I felt that an obvious breach from WP rules should be reported. Do you have a suggestion how to proceed from here forth?

Looking over Wikipedia:RCU, it seems that there is not sufficient jusjification at this point for CU. If you have more experiance with this and think otherwise, please advise (or submit a CU yourself, if you don't mind and think this will be beneficial).

I can see the reasoning for the emphasis on WP:V and WP:RS rather than truth. However, in mathematics we are in a somewhat different situation as there is more agreement between experts with regards to the truth. It is rather embarassing when there is a statement in an article that is blatantly wrong. In any case, this is not the nature of the dispute with TE. With quite a bit of work and effort, we did come to an understanding with TE regarding every particular statement.

"It occurred to me that the subspace topology on {0} union (1,2) union {3}, as a subspace of the real line, cannot be induced from an order topology." I agree. I'll be happy to discuss your other questions too. Oded (talk) 00:05, 5 July 2008 (UTC)

I think you reacted reasonably. For the checkuser, I think the "throwaway" aspect prevents it from being cut and dry, so probably not worth pursuing. To proceed with the sock case, I think it may be best to let it stand as is. An admin will close it, and decide whether to block E.i.T. or let sleeping dogs lie. Hopefully this doesn't feel like a waste of time, but hopefully it also makes it clear we'd rather be writing some math. My family is in town this weekend (US holiday), but hopefully Monday I'll gather my order topology thoughts, thanks for confirming my intuition. JackSchmidt (talk) 05:05, 5 July 2008 (UTC)

Recent exchanges make me believe I can help you with proceeding. I apply for a block on the sock page. There are chances that it stops there and we can move on. BTW, I post a text on my talk (an answer to Jack), if you're bored you may go there. Kind of last word, say, of a general nature (further discussion is not encouraged, though, since I'm not sure I'll be able to respond). Expert in topology (talk) 12:24, 5 July 2008 (UTC)

Dear Jack,

In fact the subspace topology on (1,2) U {3} cannot be induced from an order topology.

Thanks

Topology Expert (talk) 03:49, 5 July 2008 (UTC)

Thanks this is simpler. My idea is that (1,2) in the subspace topology has only two compatible orders, and they are isomorphic. Then some argument about connected components going to connected components, and defining connected component for an order somehow. Then isolated points have to behave like the integers between connected components. Tons of details to check, so I'll post a reasonable version after the weekend. JackSchmidt (talk) 05:05, 5 July 2008 (UTC)

Perfect space

Dear Oded,

Thanks for reviewing the article on perfect space. Could you please tell me what my mistakes were? Don't be reluctant to change the article on perfect space if you are certain that I have made a mistake but please inform me of the mistake beforehand.

Thanks

Topology Expert (talk) 02:57, 5 July 2008 (UTC)

The normal procedure is modify and then explain if necessary (I believe). But for now I'll just explain and not modify. I'll do this where it belongs, on the article's talk page. Oded (talk) 05:22, 5 July 2008 (UTC)

Help

Hello. Sorry to bother you, but I'm having a slight problem with User:Arcfrk on the talk page of Differential geometry of surfaces. He has attacked all the parts I added and claims quite inaccurately that I have been following him around. I have filed a report on his second unprompted attack on the talk page of wikiproject mathematics. The material I wrote has not been changed since I wrote it, but it was only a first draft. I wonder whether you could shine any light on this situation. At present I've just started writing Plancherel theorem for spherical functions. User:R.e.b. OKd the preliminary article on Zonal spherical functions. I am not sure why Arcfrk has decided to attack me in this way. Many thanks, Mathsci (talk) 23:43, 5 July 2008 (UTC)

Hello again. Thanks for posting to WPM and reassuring me about the content. I lectured some of this material recently in a seminar and have lectured harder versions of the material in the UK in courses on the Atiyah-Singer index theorem; in the UK this material was normally taught to 3rd year undergraduates in roughly this form. I would be happy with any attempt to simplify the material or make it more accessible to a wider audience, possibly with pictures. There are some nice pictures in Pelham Wilson's book and, if I have time, I might try to draw something with xfig, but I am not an expert.

On WPM I have detailed the personal attack Arcfrk placed on his user page (removed now) and his own statement that he was "withdrawing from WP editing". In view of this personal attack and because the content I added passes muster, I will continue with my edits disregarding any similar outbursts in the future. Many thanks for your help. Mathsci (talk) 06:56, 6 July 2008 (UTC)

Perfect space discussion

Dear Oded,

Please see my comments on the discussion page of perfect space. Could you also please tell me why you think that theorem 1 (in perfect space) follows from the Baire category theorem. The proof of the Baire category theorem is similar to the proof of theorem 1 but no theorem is dependent on the other.

Thanks

Topology Expert (talk) 04:41, 6 July 2008 (UTC)

Mediation

Dear Oded, thank you for responding to my request at the Differential geometry of surfaces. I also very much appreciate your fair assessment of the situation and attempts at mediation! For my part, I can promise not to initiate any personal discussions and to keep the matters civil. Moreover, this would give me a chance to resume editing mathematics articles (since you said you weren't very interested in the "past baggage", I'll spare you the details). Unfortunately, given his latest posting at WT:WPM, I have doubts that MathSci is prepared to honor a straightforward deal that you have suggested. Best regards, Arcfrk (talk) 22:12, 7 July 2008 (UTC)

Well, if this is a way to "completely disregard past disputes, treat Arcfrk politely, refrain from bad mouting him/her and treat his/her edits without prejudice" then I am Elvis Pressley. Arcfrk (talk) 06:48, 9 July 2008 (UTC)

Essential Range

Dear Oded,

Do you think that there should be an article on the 'essential range' of a function? I think that I should create one but generally it is quite difficult to give references to such a concept.

Thanks

Topology Expert (talk) 05:46, 9 July 2008 (UTC)

What is the essential range? Oded (talk) 06:25, 9 July 2008 (UTC)

Dear Oded,

The essential range of a function is defined mathematically as follows (note that: the essential range is only defined for functions that are essentially bounded and measurable on a measure space X):

Essential range of f = {complex numbers z | m ({x: abs (f(x) - w) < e) is greater than 0 for all e > 0}

note that: 'm' is the measure defined on the domain of the function

I hope that I have clearly worded this but just in case, one can describe the essential range of a function as follows:

The essential range of a complex valued function is the set of all complex numbers w such that the inverse image of each epsilon-neighbourhood of w under f has positive measure.

Actually, I only learnt about the essential range of a function fairly recently. It is one of those concepts (similar to supercompactness) that have some importance in mathematics but are not in general significant. I intuitively think of the essential range of a function as the 'non-negligible' range. I can give a few properties of the essential range later but I know for a fact that the essential range of a function is always compact as a subset of the complex plane.

Thanks

Topology Expert (talk) 05:33, 10 July 2008 (UTC)

Dear Oded,

I think that I am going to create the article after I have found a decent reference. Of course, I would appreciate it if you could give me your opinion on the matter beforehand.

Thanks

Topology Expert (talk) 05:55, 10 July 2008 (UTC)

It would be a good idea to have such an article, but I think you should not write more about the essential range than what you can find and quote from books and articles. Oded (talk) 15:29, 10 July 2008 (UTC)

Dear Oded,

Thanks for your opinion; actually I did create the article on the essential range. Note that, the book from which I found this definition actually gave an exercise which asked to prove that the essential range is compact, and asked what relation exists between the essential range and the essential supremum of a function. Basically I gave a proof that the essential range is compact and stated the relationship between the two concepts in the article. Do you think that this is OK (it is not original research since I basically gave the solution to an exercise). I also added that every bounded function is essentially bounded (which is an obvious fact (in my opinion)). So, I think that what I have written is OK since it is not original research. Only verification is a problem but since I have given a proof of each statement I think that this is also OK. If you notice, I also added a reference to the book which I got this information from. Since this book is quite large, do you think that it is necessary to include the page number of the definition or do you think people can find it through the index and it is not necessary to include the page number?

Thanks

Topology Expert (talk) 00:58, 11 July 2008 (UTC)

I think that the article you created has way too much in it. What you write as theorems are not theorems but essentially trivial excercises. I don't see why you restrict the article to essentially bounded functions. If Rudin used this concept at some point and found it convenient to have these assumptions for some particular purpose, this does not mean that WP needs an article about it. If it shows up in several books or papers, then it may deserve an article. Nothing in the article explains why this concept is important and of interest. Oded (talk) 03:56, 11 July 2008 (UTC)

Civility non-problem

Could you take a look at Talk:Separable space#Mistake? and User:Plclark the page? Both pages strike me as being contrary to the wikipedia culture, and the talk page especially seems incredibly condescending in my opinion. Thanks, JackSchmidt (talk) 13:23, 9 July 2008 (UTC)

I had a look. I agree and also Plclark seems to agree that his/her response was rude. I did not read all of Plclark's comments, but those that I did read made sense to me mathematically. Is there any particular mathematical issue that you'd like me to try and sort out? With regards to an example of a point that is in the closure of a set $A$ but not a limit of any sequence from $A$, you could take the function that is zero on the rationals and 1 on the irrationals and the dense set $A$ being the union of the $F_{n}$ mentioned in the article. Oded (talk) 16:18, 9 July 2008 (UTC)

Thanks for checking into the civility. I had tried to keep redirecting the conversation in a positive manner, but the last bit indicated he may not understand that what he was doing was bad for the encyclopedia. BTW, notice the old WP:V versus truth; I've got no idea about any of this topology junk, but I can at least point out clearly what Sierpinski, Munkres, and Willard say about it.

For the math part, basically I want to add an example to the article, but all the examples I found were not Hausdorff or were confusing to me. I found the F_n thing pretty confusing (but I think I found a reference for it, in case someone likes it and wants to avoid WP:OR). Can you check if this example is good?

Let X be the order topology on the (first? or) second uncountable ordinal. Then X is Hausdorff (I hope?), separable (how can this be?), and the last point is not a limit of any sequence, but is a limit point of some countable subset (which one?). It is sort of from the first countable article, but it doesn't seem obvious to me it works. Can it be made to work if it does not?

Basically, we want an "easy", "short" example of a Hsdf. sep. not-1st-count. space, a countable subset, and a limit point of the subset that is not a limit of any sequence form the subset. Your example is perfect other than the text explaining the F_n is not short (either in article, or in reliable source), and so inherits not being easy.

If we have an article on the example space (we have one on most of the "counterexamples in topology"), then it might make more sense to explain part of it there, and only include a shorter reference in the separable space article. JackSchmidt (talk) 16:35, 9 July 2008 (UTC)

You cannot be serious saying that you have no idea about any of this topology "junk". If you had no idea at all, then certainly you would not be able to take some facts from books and change the wording while not changing the meaning. Indeed, your questions indicate that you do have an idea of what is going on.

I'm not happy with the example that you suggest (i.e., its correctness). Hausdorff is certainly ok. As far as I can see it is not separable. Besides, I think that for many readers an example such as $[0,1]^{[0,1]}$ is more meaningful than one with ordinals. You may want to look at Kelley's book for a good illustrative example. (I don't have it at home, so cannot look right now.)

By the way, since the set of limits of sequences from a countable set has cardinality at most $2^{\aleph _{0}}$ and the space $\mathbb {R} ^{\mathbb {R} }$ has larger cardinality, it is clear that "most" points in the space would not be limits of sequences from the countable dense subset. So this provides an example without specifying a specific point. Oded (talk) 17:05, 9 July 2008 (UTC)

Thanks. Kelley p76, problem B is my ordinal example (with no claims that it is separable). However, on p77 he gives the Arens–Fort space as an example that is Hausdorff, and there is a sequence with a limit point (as a countable set), but no subsequential limits (which if I understand infinity right today, means no sequence form the set converges). I think the closure of the sequence should be Hausdorff, separable, and have a countable subset with a limit point, that is not a subsequential limit (from that subset). I'll look over this example, and then try R^R or I^I (they should be about the same for this, right? some refs use one and some the other). JackSchmidt (talk) 18:20, 9 July 2008 (UTC)

Woo, even got a reliable source now: Steen and Seebach, p54, 26.3 X-{(0,0)} has no sequence with limit (0,0), but 26.1, X is Hausdorff, and 26.4, X is separable (which I guess is obvious, X is countable). JackSchmidt (talk) 18:24, 9 July 2008 (UTC)

I think the A-F space makes sense now. I'm not sure whether to add it before or after the restructure. It seems like the statements in the article should be sourced, but some of them could be efficiently combined. On the other hand some of the examples need to come first and be clear and concise. I had another related question:

Sierpinski's General Topology is a pretty neat book. I'm curious if I can justify one my old edit summaries: I thought Sierpinski gave a construction for embedding any Hausdorff space into a separable Hausdorff space, but maybe it is just T1. Take a space X and a countable cofinite topology space Y. The proper closed sets on X union Y are A union B where A is closed, B is proper and closed. The closure of Y is X union Y. The subspace topology and original topology on X agree. The nonempty open sets on X union Y are A union B where A is open and B is cofinite. In particular X union {} is closed, but not open (that is, {} union Y is open, but not closed). X union Y is T1 iff X is T1.

The problem seems to be that Y, as a cofinite topology, is only T1. Can we instead choose Y to be the countable Fort space, and do the construction again? Is the result an embedding of every Hausdorff space into a separable Hausdorff space? Does this "merging" of two spaces have a name? It strikes me as very much like the two point Sierpinski space. JackSchmidt (talk) 19:25, 9 July 2008 (UTC)

Apologies, I cannot answer your questions right now (have to go soon), but hope to do it some time today. I had two thoughts about our earlier discussion: one nice example might be the Stone-Cech compactification of $\mathbb {N}$. Another thought is that when you ask someone a math question and they answer rudely avoiding the question, it might be an indication that they do not know the answer and are reluctant to admit it. Oded (talk) 19:35, 9 July 2008 (UTC)

I do not recall ever seeing this A-F example, nor the example you mention from Sierpinski's book. (I am not a point-set topology expert, but do like the subject, and have had use for it from time to time.) These are nice examples. If we believe the statement from the article that gives a bound on the cardinality of a separable Hausdorff space (which I am inclined to believe), then it is not true that every Hausdorff space embeds in a separable Hausdorff space, since the cardinalities of Hausdorff spaces are unbounded. The trouble with your example with the Fort space is that the proposed collection of "open sets" does not form a topology, since it is not closed under finite intersections. Perhaps for this reason the "merging" does not have a name. Its scope is not so general. For it to work you will need Y to have the property that the intersection of any two open sets is nonempty, and this fails for Hausdorff spaces (except for trivial cases where the space has one point or no points). I hope this helps. Oded (talk) 22:32, 9 July 2008 (UTC)

Wow, perfect! You exactly pinpointed the problem, that shows this specific construction cannot produce Hausdorff spaces, and showed me how to use the cardinality fact to answer an interesting question (that no, no such embedding can exist with any construction). My introduction to math was in general topology, but it has been a decade since I've enjoyed working with it. Thanks for making it look fun again. JackSchmidt (talk) 00:43, 10 July 2008 (UTC)

Just to point out a few things (unrelated to this argument), there are a few nice examples of spaces that are Hausdorff, separable but do not have the property that a point in the closure of a set is the limit of a sequence of points belonging to that set. For example, consider R^ω in the box topology. This space has a countable dense subset and is Hausdorff but does not satisfy the desired condition. A proof is as follows:

Take the set X = {(x_i) | x_i > 0 for all i}; note that 0 is a limit point of this set. Suppose there is a sequence of points in X converging to 0. Let (s_n) be that sequence and let the nth term, s_n = (s_1n, s_2n, s_3n…..). Then choose the basis element about 0 to be:

B = (-s₁₁, s₁₁) X ……X (-s_nn, s_nn)…..

Then B cannot contain any term of the sequence (s_n) since the nth co-ordinate of this term doesn’t belong to (-s_nn, s_nn).

Q.E.D

Actually, if a space is first countable, then if a point belongs to the closure of a particular set, it must be the limit of a sequence of points in that set.

Topology Expert (talk) 04:25, 10 July 2008 (UTC)

Just dropping a line to confirm that incivility was never my intent. As I said, I thought that JackSchmidt was asking the same question several times and not paying attention to my answers, but it eventually came out that the two of us were not being clear enough with each other about exactly which hypotheses were in force: i.e., it was all some kind of misunderstanding.

Related comments: (i) I'm not sure what on my user page seemed counterproductive and unwikipedic, but if it can be viewed as such then, no problem...it's gone. All it says now is that I am a mathematician and what my research specialities are. (ii) I think it is now clear that I was not being rude as an attempt to cover up a lack of knowledge. (iii) Like Prof. Schramm, I do not buy into the "Aw shucks, I'm just folks" proclamations that JackSchmidt makes sometimes: your user page says you are a mathematician and that you have a master's degree in mathematics: that's not consistent with calling topology a bunch of mumbo-jumbo! Also your edits clearly indicate that you know what you're doing. (I disagree that any random person could pick up a topology book and use it to make good edits to wikipedia articles on relatively technical topological matters. Moreover, why would they want to?) In fact, despite the fact that we seem to have had an unfortunate misunderstanding, we have already worked together to significantly improve two different articles. I look forward to further positive collaborations in the future. Plclark (talk) 02:51, 11 July 2008 (UTC)Plclark

Right. I long since regarded this incivility episode as forgotten and irrelevant. With regards to what your user page said, I actually did not see a serious problem there. Perhaps there was one or two statements that were blunt (I don't actually remember, and there's no need to look at the diffs and check now). But, on the other hand, it did indicate your frank perspective and some statement of your WP style, which is valuable. From what I have seen so far, I agree that there is very significant variability in the quality of articles. Oded (talk) 03:45, 11 July 2008 (UTC)

Yup, all good now as far as I am concerned. Plclark immediately addressed my concern on this isolated misunderstanding (and, as he said, everything else between us has been hunky dory), and I apologized for my own misunderstanding. Oded or I probably should have changed the heading here after Oded's first response, since the conversation after that was almost entirely topology!

I actually asked Oded to look at it, because he'd recently been accused of being condescending (when in fact he was trying to be quite a nice person). I figured he was the perfect person to tell me clearly that Plclark was a reasonable guy, and that it was fine to move on.

Lately I've been trying to help out the wikiproject math community a bit more, and have been practicing getting some articles to FA status. This means I'm editing just as far out of my expertise as when I'm typo checking some 18th century buccaneer stub, but now people might get confused and think I know the difference between a gunnel and a jackline, or a first countable and separable space. JackSchmidt (talk) 04:39, 11 July 2008 (UTC)

Essential Range (importance)

Dear Oded,

I will respond to all your comments in dot points:

• The reason why one only considers functions of class L^infinity (μ) [i.e essentially bounded functions] is explained through the following comments:

1. The essential range of an essentially bounded function is always compact.

2. Generally, essential range of an essentially bounded function always lies within a closed ball in R² of radius equal to the essential supremum of that function.

Note that, the properies I have described above are not necessarily reasons why one must restrict to essentially bounded functions in the definition. One could add the hypothesis in theorem 1 that the essential range of a function is compact if the function is essentially bounded. I think that the essential range is only defined for essentially bounded functions mainly because if the function is not essentially bounded, its essential range may not satisfy some important properties. I can list down some properties of the essential range of an essentially bounded that do not necessarily hold if the hypothesis of essential boundedness is ommitted.

• You said that the essential range is not a very important concept. It is certainly not crucial to its field but it does have applications in cohomology theory.

Note that I am not an expert on the essential range but having done some research on it, I know that it is a concept in mathematics and does have some importance. I thought of some properties of the essential range and I can conclude that it does have some role in mathematics. If you analyse the defintion more carefully, perhaps you will understand why it may be of some importance. Anyway, I don't see what is wrong about having an article on the essential range since it is a concept in mathematics (like any other concept; for example Borel set).

Thanks

Topology Expert (talk) 05:36, 11 July 2008 (UTC)

I am not saying that it is unimportant. All I said is that there is noting in the article that explains the importance, and that is a very serious shortcoming of the article. My guess is that the concept of essential range comes up as a tool to prove some Theorems, but not theorems about essential range. For example, the fundamental group is a concept that has many uses. It can be used to show that the sphere and the torus are not homeomorphic, for example. The concept of a Borel set is natural because the Borel sigma-field is the smallest sigma field that contains open sets, and both the concept of the sigma-field and of an open set are important. You quote Rudin. I am sure that Rudin does not mention the essential range as a fun concept by itself, but as a tool for some other objective. Wouldn't it be a good idea to mention that objective as an application of essential range in the article? Oded (talk) 05:46, 11 July 2008 (UTC)

Dear Oded,

Sometimes in textbooks, 'new' definitions are provided for the purpose of providing an exercise to the reader. These 'new' definitions may not actually be analysed in the textbook. I think that this is the same for Rudin's book. I have seen the index and the only reference to the essential range is given in the particular exercise I mentioned. I agree that it would be a good idea to mention an application of the essential range but currently I can't since I have no references apart from Rudin's book. Perhaps I will have to find one but what I have written so far is better than nothing.

Regarding the examples you have given (borel sigma field and the fundamental group), they are very important concepts in mathematics and it would take one minute of searching to find an application of these concepts. Pretend you just learnt what metacompactness means; then it would take a lot more time to find applications of this concept. The same thing is true for the essential range of a function (in my opinion). Note that I completely agree with what you say, however.

Thanks

Topology Expert (talk) 09:30, 11 July 2008 (UTC)

Of course, WP has much room for concepts much less important than the fundamental group. But I don't think that being defined as part of an excercise in a textbook is sufficient reason for a concept to have an article on WP. Possibly, in this case there are other reasons. But if we don't find other more significant mentions of this in the literature, then the article should be deleted. Oded (talk) 16:13, 11 July 2008 (UTC)

Dear Oded,

Do you agree that the essential supremum is an important concept in mathematics (when I say 'important' I mean worth studying)? If so, then I can list quite a few relations between the essential range of a function and the essential supremum. In fact, the essential range also has a relationship with another concept:

Let Z = L²(-infinity, + infinity) and let f be a function in Z. Let M be the multiplicative operator taking and function g in Z to fg. Then the spectrum of M is equal to the essential range of f.

I found out this relationship after 10 minutes of research. I will probably find out more such relationships in the future. All I am suggesting is that the essential range must be of some importance and I can find evidence to prove this.

Thanks

Topology Expert (talk) 04:36, 12 July 2008 (UTC)

But, of course, you know that you should not put your own research on WP. Oded (talk) 01:30, 13 July 2008 (UTC)

Dear Oded and Topo,

Please see my comments on Talk:Essential range. Just now I did a MathSciNet search for essential range: 73 articles, from a 1959 Annals paper to 2007. For comparison, searching for "Ruth" and "Aaron" -- c.f. Ruth-Aaron pair -- on MathSciNet gives 3 articles; searching for "McGwire", "Maris", "Sosa" -- c.f. Maris-McGwire-Sosa pair -- gives none. The latter article was nominated for deletion and the result was "keep". I hope this gives some perspective on how un/important a math article can be and still be kept.

The fact that Topo mentioned about the spectrum is eminently sourceable: I gave a reference on the talk page. I would like to think that by "research", Topo means this sort of "library research". I have already expressed on his talk page my concerns about his unsourced verifications and am hoping that he will respond positively to these concerns. Plclark (talk) 07:21, 13 July 2008 (UTC)Plclark

• Right. My remark on the talk page was meant to be verbatum. All I was saying is that the article does not explain or motivate sufficiently the importance of the notion. I was not implying that the notion is unimportant (nor that it is important). I would say that based on the present content of the article and the comment by Topology Expert that there is an excercise in Rudin about it, I would not think that the article is worth keeping. Based on your literature searches, I am ready to concur that the article should be kept.
• I believe it is WP policy that in general comparison should not be an argument for deciding whether a subject deserves a WP article or not. That is to say, arguments such as "since this notion is included, we must also include a more important other notion" or the converse are not deemed to carry weight.
• Plclark: I notice you have done quite a bit of very good work on several articles. I want to congratulate you for making such a positive contribution.
• TopoExpert: I take this opportunity to elaborate on a point I was making in the talk page of essential range (and one which also I believe Jack made). Jack and I were saying that the essential point in what is now theorem 2 (and what I think is not sufficiently important to be called a "theorem") is that the essential range is closed. Right, this does not imply compactness and compactness does not hold of we do not assume that the function is in $L^{\infty }$. However, it is very common knowledge among people who know what compactness is that closed and bounded sets in $\mathbb {R} ^{d}$ are compact. The fact that the essential range is closed is very general. It holds also if the range of the function is an infinite dimensional Banach space, for example. From this, the compactness when the function is essentially bounded is an immediate corolary (one that is so obvious, it does not require any mention).

Oded (talk) 08:52, 13 July 2008 (UTC)

Dear Oded,

Then add a corollary to theorem 2 stating that the essential range of an arbitrary function is always closed. However, I think that you should still keep theorem 2.

Thanks

Topology Expert (talk) 03:16, 14 July 2008 (UTC)

I presently don't have any plans to rewrite this article, since I have not seen what the literature says about it. If it was up to me, I would define the essential range as the support of the pushforward measure, because that is what it is. Then the user can build on his knowledge of these concepts to deduce properties of the essential range. (Woe the user who learns about the essential range before learning these more important concepts.) But it is not really up to me to decide about this. Oded (talk) 07:23, 14 July 2008 (UTC)

Second moment

I reverted your changes to Second moment. That isn't really an appropriate case for a dab page. Firstly, there are only two entries. When there are only two ambiguous terms, the preferred approach is to just put dablinks at the top of each of the articles, pointing to the other. I added one to the top of moment (mathematics). Additionally "second moment method" is not actually ambiguous with "second moment". See WP:disambiguation and WP:MOSDAB for more information on how to construct disambiguation pages. --Srleffler (talk) 22:49, 13 July 2008 (UTC)

I noticed, but thanks for contacting me. I certainly see the logic in what you are saying. But somehow it does not make sense to me that second moment would redirect to moment. Why would anyone link "second moment", rather than "second moment"? If anyone does that, my guess is that they really want something else. But in any case, it is ok by me to leave things as they are now, if you think that's best. Oded (talk) 22:53, 13 July 2008 (UTC)

Actually, to my eye "second moment" would be the better link. It's best to link the most specific term, and allow the reader to locate the more general topic through links in the target article, or through a redirect as in this case. This is actually specified by Wikipedia guidelines. The Manual of Style says:

Links should use the most precise target that arises in the context, even where that is merely a simple redirect to a less specific page title. Do not use a piped link to avoid otherwise legitimate redirect targets that fit well within the scope of the text. This assists in determining when a significant number of references to redirected links warrant more detailed articles.

--Srleffler (talk) 02:12, 14 July 2008 (UTC)

Analysis fun at refdesk

Thanks for confirming an analytic function can be that weird on the boundary. I knew it could be unbounded on a dense set, but was nervous just taking 1/f; your exp(-Re(f)) was very clever.

Is there a simple way of stating this in terms of PDE? Like "I have some continuous boundary data, so there is a continuous function on the closed disk that agrees with the boundary data on the unit circle and is harmonic on the open unit disk." I think there is just a formula for the solution in terms of the boundary data on the disk.

Yes. This does work. This was my initial intuition as well, but I think I got confused a little bit (never mind by what) and lead to a longer solution. You can take a function on the circle that takes values in $[0,\infty ]$ (infinity intentially included), such that the function is continuous at the places where the value is infinite, and the integral of the function with respect to arclength is finite. The harmonic extension of this function (which can be described using the Poisson kernel) can serve as the harmonic function in that construction. I guess that's what you meant, but I thought I would make it explicit.

Thanks for pointing out arclength is not continuous. This strikes me as a really motivating example for sobolev spaces (or hoelder spaces). You can't expect control of the derivatives without requiring some control on the derivatives! I posted a sort of discussion of this on the refdesk, and found a fairly cool example, modulo proving that a certain integral was constant.

I just took the obvious example of a wiggly horizontal line (the sine wave), but both maple and matlab agree the arclength of my guy is constant in n (about bigger than the arclength of the straight line of course)! I thought that was pretty wicked, but I wanted to make sure. Maple makes a habit out of misevaluating integrals, and the matlab method was not very specific on its true hypotheses (I used the oscillatory quadrature method, and its error bounds appeared consistent, so I suspect my function is ok).

It would be nice to find a "Thomas" or "Stewart" integral that came from such an example (that is, a first year engineer could be assigned the problem of evaluating the integral, and could be expected to see that the sequence of curves clearly approached a straight line in the limit, but the arclengths did not approach the right number). Making the integrals diverge to infinity is pretty obvious, but I wanted to be a little more subtle than that. JackSchmidt (talk) 20:02, 14 July 2008 (UTC)

RE: Topology Expert vs. Expert in topology sock case

User:Expert in topology overall seems to have a better grasp of wiki-syntax than User:Topology Expert (although I'm sure the latter will pick things up). The best illustration of this is probably any of their edits to discussion pages: User:Expert in topology writes like most people, using a single suitably-indented paragraph with his signature at the end. User:Topology Expert hasn't got the hang of this yet, and writes in an unindented letter-like style complete with salutation and valediction, with his signature in the latter. While I'm sure it's possible to fake such a thing, it would be at least a bit awkward, and it seems like if you were going to try and maintain that the accounts were separate that you'd pick a different name in the first place! Based on all this, I'm inclined to believe User:Expert in topology's own story regarding his origins, hence User:Topology Expert would just be a (by now slightly confused) spectator. --_{tiny plastic} Grey Knight ⊖ 08:44, 16 July 2008 (UTC)

The first real edit of User:Expert in topology (on this page) has both a salutation and a valediction and has two paragraphs. It therefore seems that your observation seems to point the other way completely. It is possible that EiT later adapted, when it was explained to "them" that this is not legitimate behaviour. So it is likely that at the begining, there was no intention to completely decieve. Oded (talk) 17:53, 16 July 2008 (UTC)

Sorry, thought I'd replied to this earlier... I saw the similarity in that post, but dismissed it as a coincidence since it doesn't conform to User:Topology Expert's usual style. No change in my opinion I'm afraid! :-) --_{tiny plastic} Grey Knight ⊖ 15:10, 17 July 2008 (UTC)

I've proposed leaving the case for janitor attention for the time being, since it's been open for over two weeks now. Please comment on the case page if you think that's a good idea. --_{tiny plastic} Grey Knight ⊖ 15:31, 17 July 2008 (UTC)

Oh, and I asked User:Topology Expert to strike that last comment of his, I think he was just upset at the time of posting (never a good idea to post while tired and emotional!). --_{tiny plastic} Grey Knight ⊖ 15:33, 17 July 2008 (UTC)

User:Topology Expert seemed to be under the impression that the original problems you guys were having were because of mistakes in articles; I sent him this note explaining why it's OK to make mistakes in a wiki (as long as you put in reasonable effort), and I said I'd mention it here too. The whole "verifiability not truth" thing is really hard to get a handle on with respect to mathematics articles, for obvious reasons which I'm sure it would be rude of me to point out to your good self! :-) (I see Wikipedia talk:WikiProject Mathematics/Proofs is still being talked about) Anyway, hope all is well. --_{tiny plastic} Grey Knight ⊖ 08:04, 18 July 2008 (UTC)

Thanks for your efforts in this matter. Oded (talk) 17:05, 18 July 2008 (UTC)

Paracompactness

Dear Oded,

I think that the section, 'examples and counterexamples', in the page on paracompactness, gives the wrong impression about paracompactness. I hope that you agree with me and the following reasons that I put forth:

I agree that the paragraph

As you might guess from the generality of most of the examples above, it is actually harder to think of spaces that are not paracompact than to think of spaces that are. The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.) Another counterexample is a product of uncountably many copies of an infinite discrete space.

should be changed and merged into the list above it without its first sentence. Also, "most famous" should be dropped.

• It is not 'harder' to think of spaces that are paracompact than to think of spaces that are paracompact. For instance, if a Hausdorff space is not normal (or regular) then it cannot be paracompact. This automatically eliminates 'many' spaces from being paracompact.

• I think that the general reader may not know enough about the long line (I think that most textbooks in topology do not actually mention the long line!). Therefore, by using the long line as a counterexample, the reader gets the impression that very 'complex' spaces serve as counterexamples. This is not so.

• Also, there are many more 'common' spaces that are not paracompact. In particular:

• K-topology
• Sorgenfrey plane
• Moore plane (although this is not common)
• Particular point topology
• The one-point compactification of a uncountable discrete space

The last one is compact and hence paracompact. I did not check the others. I think the long line is more commonly known than any of the first 4 examples that you list.

I could probably list a lot more. Therefore, I think that this section should be re-written so that both examples of paracompact spaces and non-paracompact spaces are provided.

In fact, why include a section entitled, 'definitions and relevant terms'. As far as I know about Wikipedia conventions (and I don't think I know much), the reader should refer to another article if he/she doesn't understand the terms in one particular article.

I disagree. I think it is accepted that in some cases relevant definitions that are essential to the content of an article are given within the article even if they have their own articles.

Could you please have a look at the reasons that I have put forth? If anything, some parts of the article should be slightly modified (in my opinion).

Thanks

Topology Expert (talk) 14:46, 23 July 2008 (UTC)

see my comments within yours. By the way, can you explain why the long line is not paracompact and why $\mathbb {Z} ^{\mathbb {R} }$ is not paracompact? Oded (talk) 17:13, 23 July 2008 (UTC)

As for the first, it is a Hausdorff manifold but is not second countable (no countable basis). For Hausdorff manifolds, second countability is equivalent to paracompactness. And it's pretty easy to see from the definition of long line that it is not second countable. --C S (talk) 17:22, 23 July 2008 (UTC)

If you take an uncountable disjoint union of $\mathbb {R}$, isn't that a paracompact Hausdorff manifold that is not second countable? It seems to me that it is, but I might be confused. Oded (talk) 17:40, 23 July 2008 (UTC)

To help clarify the misunderstanding, here is the reason why I think that an uncountable disjoint union of $\mathbb {R}$ is paracompact. Suppose you have any cover. Intersect every element of the cover by every copy of $\mathbb {R}$. This gives an open refinement. Now since each $\mathbb {R}$ is paracompact, the sets in this refined cover that intersect that copy of $\mathbb {R}$ have a locally finite open refinement. The union of these would form an open refinement of the original space. No? Oded (talk) 17:44, 23 July 2008 (UTC)

Yes. In general, any direct sum (a.k.a. disjoint union) of paracompact spaces is paracompact. C S forgot the hypothesis that the manifold must be connected; in fact, that it have only countably many connected components is sufficient. Since in "real life" one never meets manifolds with uncountably many connected components, it is easy to forget to explicitly mention this. Of course the long line is path-connected. There is also a very striking argument for its nonparacompactness in terms of the fact that it is a one-dimensional manifold whose tangent bundle admits a nowhere vanishing section but is nevertheless not trivial as a vector bundle (!!). If I remember correctly, this can be found in an appendix of Michael Spivak's Introduction to Differential Geometry, Volume I. It might be nice to add this to the article, which also deserves to have more of the applications of paracompactness in geometry and topology, especially as regards fiber bundles and classifying spaces. Plclark (talk) 17:57, 23 July 2008 (UTC)Plclark

Yup, paracompactness has lots of applications in differential geometry and algebraic topology. The article is too restricted and only relates paracompactness with the concepts of point-set topology.

Topology Expert (talk) 06:58, 25 July 2008 (UTC)

Good catch, both of you! --C S (talk) 21:38, 23 July 2008 (UTC)

In fact, X = $\mathbb {Z} ₊^{\mathbb {R} }$ is not paracompact since it is Hausdorff but not normal. I think that the 'easiest' way of proving that a space is not paracompact is to show that it is Hausdorff but not normal. Here is a proof (treat X as a function space)

Let A_1 be the set of all functions f: R -> Z₊ such that for each i not equal to 1, f^(-1) (i) contains at most one element. Note that A_1 is closed in X. Similarly, defined A_2 to be the set of all functions g: R -> Z₊, such that g^(-1) (i) contains at most one element for each i not equal to 2. Note that A_2 is not closed in X. Note also that A_1 and A_2 are disjoint.

I don't have time to do this, but one can prove that there are no disjoint neighbourhoods about A_1 and A_2 so that X is not normal. Clearly, X is Hausdorff (being the product of Hausdorff spaces). Therefore, X cannot be paracompact. If necessary I can supply a proof later.

Topology Expert (talk) 07:40, 24 July 2008 (UTC)

Also, note that the first uncountable ordinal is not paracompact. If L is the long line, and if w_1 is the first uncountable ordinal, then w_1 X {0} is homeomorphic to w_1 and therefore not paracompact. However, w_1 X {0} is a closed subspace of L; since it is not paracompact, L cannot be paracompact.

Q.E.D

Topology Expert (talk) 07:40, 24 July 2008 (UTC)

By the way, the first uncountable ordinal (w_1) is not paracompact since:

• The product of a paracompact space and a compact space is paracompact (by the tube lemma)

• If (w_1) were paracompact then the (w_1) X (Closure (w_1)) would be paracompact; a contradiction since this space is Hausdorff but not normal

Topology Expert (talk) 10:56, 24 July 2008 (UTC)

Thanks for the explanations. I'll think about it. Oded (talk) 05:03, 25 July 2008 (UTC)

Moore Space

Dear Oded,

There is an open problem in topology which asks whether every locally connected, locally compact, pseudocompact Moore space is metrizable. According to the definition of a Moore space in Wikipedia, a trivial counterexample to this conjecture is an arbitrary (non-empty) set given the indiscrete topology. I am led to believe that the definition of a Moore space as given by Wikipedia is incorrect since I obtained this conjecture from a reliable source which I can cite (a book).

Not only is the definition of a Moore space wrong (as I believe) but the lede is also badly structured. The article states,

According to Vickery's theorem a topological space is a Moore space iff it is regular and developable.

If this definition is a consequence of Vickery's theorem, then what is the original definition? Also, in the 'examples' section, the Sorgenfrey line is given as an example of a completely normal, heriditarily separable space that is not a Moore space. I think that this example is wrongly placed; until the reader reads the latter part of the article on Jones' theorem, he/she will not understand the significance of the example.

Also, why state in the 'examples' section that a Hedgehog space is a Moore space because it is metrizable? Every metrizable space is a Moore space so why name an obscure example of a metrizable space?

I don't know much about Moore spaces but my current understanding has led me to believe that the article should be completely rewritten. I am mainly worried about the definition since it is the main component of the article. Perhaps the definition should state that a space is a Moore space iff it is a regular, T_1 space that is also developable; so a Moore space should also satisfy the T_1 axiom. I know for a fact that according to Munkres' book, a regular space is one such that points are separated from closed sets and that one-point sets are also closed. According to Wikipedia, a regular space doesn't need to satisfy the T_1 axiom. So according to the definition in Wikipedia, the definition of a Moore space is wrong. Could you please give me your opinions on this matter?

Thanks

Topology Expert (talk) 07:25, 28 July 2008 (UTC)

I agree with all of Topo's points. Further discussion is given on the article's talk page. Plclark (talk) 08:50, 28 July 2008 (UTC)Plclark

Thanks Clark for replying. I don't know a thing about Moore spaces. Of course, in order to fix the article, you would need to find a reliable source where Moore spaces are defined.

By the way, it is my opinion that for such advanced math articles it is better to give the definition of the concept in the lede paragraph, when possible. Oded (talk) 15:16, 28 July 2008 (UTC)

Thanks!

		The Reference Desk Barnstar
		Thanks for helping me on the Mathematics Reference Desk! --Ye Olde Luke (talk) 17:23, 31 July 2008 (UTC)

In particular, you and Tango did much more than the others, so you both also recieve:

		The Guidance Barnstar
		for persistence and particularly useful help with my problem. :-) --Ye Olde Luke (talk) 17:23, 31 July 2008 (UTC)

Paracompactness and the definition of an Atlas

Dear Oded,

According to Munkres' book, 'Topology a First Course' (page 223 in the first edition), a manifold is a second countable Hausdorff space that is locally Euclidean. Wikipedia accepts spaces that are not second countable as manifolds. I do not think that this is the right convention to follow. My reason being is that manifolds defined as Hausdorff, second countable locally Euclidean spaces can be imbedded in finite dimensional Euclidean space whereas manifolds (as defined by Wikipedia) cannot. In fact, the long line is a manifold (according to Wikipedia) but cannot be imbedded in R^w if w is a countable index set. Therefore, it can only be imbedded in R^J for uncountable J. Such spaces should not be allowed as manifolds for this reason. I am reluctant to change the definition because most articles in Wikipedia follow this definition of a manifold. For instance, according to the article on paracompactness, the long line is a non-paracompact space that is a topological manifold. According to 'my' definition, all manifolds should be paracompact. Could you please give me your opinion on this?

There was a long discussion about this here (see also the section that follows). Many wikipedians contributed to the discussion. I don't think I'm qualified to add to the discussion. Oded (talk) 15:50, 11 August 2008 (UTC)

Secondly, please have a look at the page on Atlas (topology). You will probably find that it is not formal enough and seems to be directed to someone who does not know 'enough' topology. Here is my criticism of the article (I have much more criticism but these are the main points):

• The article refers to a manifold as a 'complicated space'; this word is not mathematical and doesn't explain what a manifold really is

• Additional to this, the article states that a manifold is made up of 'simple spaces'. According to this, a 'simple space' could be an open interval in R. But any open interval in R is also a manifold (being homeomorphic to R) which means that according to this, a space can be 'simple' and 'compicated' at the same time

• There is no such concept of a 'simple space' in mathematics anyhow (i.e it is not a mathematical term)

In fact, the user who wrote the majority of this article is User talk:Waltpohl (about four years ago). I contacted him but I don't think he is active on Wikipedia at the moment. From previous experience, deleting the article is not appropriate so I think that the article should be completely rewritten. Could you please give me your opinion on this?

Thanks

Topology Expert (talk) 10:01, 11 August 2008 (UTC)

You should list your criticisms on the article talk page, and so people who watch that article can reply. (It does not matter who wrote what initially.) (I replied concerning the assumptions for manifold above.) Oded (talk) 15:50, 11 August 2008 (UTC)

End (topology)

Dear Oded,

Please have a look at my recent edit to Talk:End (topology). I don't think that your characterization of an end is correct (does not agree with the definition as given by Wikipedia). Did you mean that your definition of an end is the right one as opposed to that given by Wikipedia?

Topology Expert (talk) 11:11, 11 August 2008 (UTC)

Dear Oded,

I have found requirements to make the two definitions equivalent; they are equivalent for Hausdorff, σ-locally compact spaces. Whenever I say 'your definition' I just mean the definition that you re-formulated from the article.

Proof:

Let X satisfy the given hypothesis. For each point, x in X, choose a compact subset C_x of X that contains a neighbourhood U_x of x. The collection of all {U_x} [indexed by the space X] forms an open cover of X and therefore has a countable subcover {U_xi} for i in N. Choose for each element in the subcover, an element C_xi containing it. Then X equals the union of all C_xi for i in N. Let A_n equal the union of all such C_xi for i varying from 1 to n. If e is the function that maps each A_n onto its complement, then we assert e satisfies the condition of an end. Since X is Hausdorff, each A_n is closed so that its complement is open. If V_i equals to the union of all U_xi for i varies from 1 to n, then the union of all V_i for i in N is X. Note also that the closure of the complement of each A_n is a subset of the complement of V_i which means that the intersection over all n of the closure of the complement of A_n is empty. Hence the intersection of the closures of all sets in the range of e will be empty. Therefore, e is an end as desired.

Q.E.D

For an uncountable set given the co-countable topology, there exists no function satisfying your hypothesis that is an end since there is only one such function. Whereas, in my proof I have shown the existence of a function satisfing your hypothesis that is still an end.

The domain of the function constructed in the proof is a subset of the collection of all compact sets but it still is an end. Perhaps your definition should allow functions that are defined on any subcollection of the collection of all compact subsets of X. In fact, an end is a countable collection of sets; according to your definition an end is a function defined on all compact subsets of X and therefore need not be countable.

Therefore, I am led to believe that your definition has to be modified quite significantly in order to agree with the definition given in the article. These are the possible modifications necessary:

• An end should be a function defined on a countable subcollection of the collection of all compact subsets of X

• You said that an end is a function that assigns to every compact set K a connected component e(K) of $X\setminus K$; you are assuming that the component of the complement of K is open which need not be the case because:

a) In locally connected spaces, components of open sets are open. If $X\setminus K$ is open, then X has to be locally connected to ensure that e(K) is really a neighbourhood of an end.

b) Also, whoever assumed that $X\setminus K$ is open? It is not open unless compact subsets of X are closed, i.e the Hausdorff condition must be assumed.

After these two modifications are made, your definition of an end will agree with the article's definition for σ-locally compact spaces.

I am not very knowledgeable about ends but the article says that an end may be used to 'compactify' a space. If this is true, then adding to compactness-related axioms to your definition may not be the best thing to do for the article's purposes. Do you have any opinions on this? In my opinion the definition of an end through functions should be removed since the original definition given by the article is the simplest and the most appropriate.

Thanks

Topology Expert (talk) 07:40, 12 August 2008 (UTC)

Fibre Bundle

Dear Oded,

I am worried about the article on fibre bundles since such an important topic deserves more content. For instance, there is no real explanation of how two fibre bundles can be isomorphic or what it means for a map between bundles to be a morphism. Even trivial bundles are not defined properly; a bundle is trivial iff it is isomorphic to the trivial bundle. The article however gives an ambiguous definition of what it means for a bundle to be trivial. For instance, my point is that saying that a space is R has two meanings, one meaning may not be interpreted by the reader (the topological interpretation of this). In the case of fibre bundles, saying a bundle is trivial iff it is the trivial bundle is correct, but the definition should be more formal. Mainly, the article says that a bundle is trivial iff it looks 'globally' like a product. I think that once a detailed discussion on isomorphisms is included, this should be changed. Could you please have a look at this?

The article does not also properly define the real definition of a fibre bundle; it only defines locally trivial fibrations. Fibre bundles that are important in mathematics, are defined through a structure group acting effectively on a fibre, and a collection of charts that are maximal with respect to certain axioms (that the composition of an element of a chart over an open set and the inverse of another such element over the same open set provides a homeomorphism of a fibre that is given by action of a certain element of the structure group on the fibre). Such fibre bundles are the 'real' fibre bundles and should be properly defined. In practice, some authors choose to omit some axioms that a fibre bundle is required to satisfy. In particular, the axiom that the topology on the structure group should be compatible with the fibre bundle (i.e the transition map between two charts has to be continuous relative to the topology on the structure group) is sometimes ommitted. If the article is to be improved, which definition should be included and what axioms are to be ommitted?

Also, there are so many interesting theorems regarding fibre bundles that could be included. I have not seen it being mentioned anywhere in Wikipedia that the tangent bundle over a manifold is just a fibre bundle with fibre homeomorphic to the tangent space corresponding to a particular point on the manifold. I think that this is how tangent bundles are formally defined (once someone has established the notion of a tangent space) as opposed to the definition given in tangent bundle. Therefore, I am led to believe that the article on tangent bundles should be expanded. I am ready to include the definition of an isomorphism between bundles but I am not sure how to include commutative diagrams in the article. Do you know how to do so? As you suggested, I have included some comments on this on the article's discussion page but I doubt that I will get a response for sometime. Could you please give me your opinion on the standards of the article?

Thanks

Topology Expert (talk) 07:27, 1 September 2008 (UTC)

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