Talk:Nowhere dense set

Empty interior

Note:

A subset A of a topological space X is nowhere dense if and only if its interior is empty. This definition doesn't make sense but it is in fact the real definition originally given (and still accepted today) by Baire. Therefore, I believe that the definition should be changed. For verification please see Munkres' book for his definition of a nowhere dense set.

Topology Expert (talk) 07:37, 22 June 2008 (UTC)

That definition makes no sense. Can you quote Munkres' actual definition? I don't have access to it. Oded (talk) 06:09, 23 June 2008 (UTC)

Contradiction?

From the article:

A nowhere dense set is always dense in itself.

From "dense in itself"

a subset A of a topological space is said to be dense-in-itself if A contains no isolated points.

The integers as subset of the reals are nowhere dense and contain only isolated points. Did I get something wrong, or is it the sentence from the article? 134.169.77.186 (talk) 15:18, 16 April 2009 (UTC) (ezander)

Oops, my mistake. I had linked the phrase "dense in itself" to "dense-in-itself", but this is not what was meant. A subset A of a topological space X is said to be dense-in-itself if no point of A is isolated in X. A subset A of a topological space X is said to be dense in X if the closure of A is X. X is dense in X, i.e., every topological space is "dense in itself". I will remove the link. — Tobias Bergemann (talk) 10:13, 24 April 2009 (UTC)

First, even without the link I get confused with the statement, dense in itself means the set has no isolated points. Now, after I have read the discussion here it still making no sense to me since every subset of a topological space is "dense in itself" in the relative topology. That's why the meaning of "dense in itself" has no relation to the set be dense on itself as a topological space: it is trivially satisfied. --Ricardo.correa.silva (talk) 16:48, 1 April 2017 (UTC)

Characterization in terms of boundaries should not be definition

In all the well-known sources (Bourbaki, Narici, Willard, Engelking, etc, etc), the definition of nowhere dense set is either in terms of the interior of the closure, or the set not being dense in any nonempty open set. The fact that such sets can also be characterized in terms of boundaries is given as a result (mathematical fact), but not as the definition, especially as it is not as direct and easy to check as the other definitions mentioned above. This article should follow common practice and not treat this fact about boundaries as an alternate definition. I have added a paragraph in the Properties section about these boundary related facts. Unless somebody can make a valid argument to the contrary, I will soon remove the definition in terms of boundaries. PatrickR2 (talk) 00:05, 28 July 2022 (UTC)

The planned change has now been made. PatrickR2 (talk) 05:02, 31 July 2022 (UTC)

Hi @PartickR2:, you reverted my edit and told me to "See prior discussion in the talk page from July 2022". This is that discussion, right? It seems to me that the point of the paragraph above is

"This article should follow common practice and not treat this fact about boundaries as an alternate definition."

I do not see how my edit goes against this. This is the relevant part of my reverted edit:

Boundary

An arbitrary set $S\subseteq X$ is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the exterior of $S$). A closed set is nowhere dense if and only if it is equal to its boundary, if and only if it is equal to the boundary of some open set (for example the open set can be taken as the complement of the set).

This does not treat this fact about boundaries as an alternate definition. If it does then please explain how, because I really do not see it. Mgkrupa 00:52, 12 January 2023 (UTC) Mgkrupa 00:52, 12 January 2023 (UTC)

Yes, this is the discussion in question. Sorry, I need to see all the other changes you made in your reorg before being able to comment. Will not have time now, but will comment by tomorrow. PatrickR2 (talk) 02:35, 12 January 2023 (UTC)

Here are my comments. In the (current) version without your proposed change, we have basically two sections: Definition and Properties. That seemed very clear to me: first a section explaining the equivalent ways to define the concept, then a section giving the properties. I had moved the boundary information to the Properties section for the reasons explained above. Now you moved that back into the Definition section via a subsection, giving it equal footing with some of the other characterizations, as well as moving other properties into the definition section. If we want to, we should keep the Definition section focused on the definitions, and keep the Properties section with subsections as necessary to present information, but not mix defs and properties.

Now regarding the contents of the Definition section without your change, I did not modify that, it's a left over from way back. I agree with you that it may not be optimal and we could maybe arrange that somewhat. There are various equivalent ways to characterize nowhere dense sets: not dense in any open set, closure with empty interior, dense exterior, and various others also mentioned. Actually, in many topology books it seems that the most commonly presented is closure with empty interior and not the one presented as "the simplest one" here. It really does not matter, they are all equivalent. We should give the various characterizations and explain why they are equivalent, and the current Definition section does that, but it does not flow well. In any case, the boundary information belongs to properties and not the definition I think. Or do you disagree with the reasons at the top and have a specific reason to give equal footing to the boundary characterization as part of an alternate definition?

I also had other comments about the rest of your reorg, but let's start with this. PatrickR2 (talk) 00:48, 13 January 2023 (UTC)

Unnecessary proof

Recently someone thought of adding a proof of the fact that the union of a finite number of nowhere dense sets is nowhere dense. The issue of having proofs in wikipedia articles has been discussed multiple times on Wikipedia:WikiProject Mathematics and. See in particular Wikipedia:WikiProject Mathematics/Proofs and other guidelines, in particular WP:NOTTEXTBOOK. Quoting a few passages from that: It can be helpful to think of Wikipedia's mathematics articles as akin to survey articles in mathematics journals. ... Unlike textbooks, Wikipedia does not strive to provide an axiomatic introduction to mathematics. Thus it is not necessary for a Wikipedia article to prove every fact that is mentioned. Our best articles include references to good textbooks that the interested reader can consult for an axiomatic presentation of the field.

So basically, it is misguided to provide proofs in an article without a serious justification for doing so (unless the article is specifically about the proof itself of a theorem). Like it says in the survey, an article works best when it provides links for textbooks where an interested reader can dig deeper to further their understanding. In this particular case, there are already two references for that fact, and one could add more. One could also ask, why prove this particular fact and not others, maybe more important ones in the article? (And starting to pepper the article with a bunch of proofs will surely not help the encyclopedic goal here.) Furthermore, the proof that was presented is certainly not an optimal one. It is overly verbose, it makes a reference to ProofWiki, which is not an acceptable source in wikipedia (ProofWiki is full of errors, woefully incomplete in general, poorly presented, and most of all, it is considered "user generated content" WP:UGC). And there are simpler and cleaner approaches to prove this fact. Seems the editor here tried to "do his own research" (see WP:NOR). For these reasons, I am removing this proof. PatrickR2 (talk) 05:00, 29 May 2024 (UTC)