Talk:General topology

Merge from point-set topology

I see that an ambitious editor has recently been writing a detailed article on point-set topology. At a glance, it looks like nice work (I haven't checked in any detail).

But I don't think that most workers make any serious distinction between point-set topology and general topology. I suppose general topology could also include pointfree topology, but that's a fairly minor sidelight IMHO.

So to recap, I think the new material is great, or at least is probably great (as I say, I haven't really checked), but I think it ought to be here, not there. I don't see any great value in having two articles on this topic. --Trovatore (talk) 02:58, 30 November 2013 (UTC)

Point-set topology was disambiguated because the term 'general topology' as used on Arxiv (http://front.math.ucdavis.edu/math.GN) and other research math websites is much broader than just point-set topology. There are conferences with sessions dedicated to general topology, such as the Spring Topology and Dynamics Conference in Connecticut; topics this year included compactifications and continuum theory. I will follow the consensus.Brirush (talk) 03:07, 30 November 2013 (UTC)

Why are compactifications and continuum theory not part of point-set topology? --Trovatore (talk) 03:45, 30 November 2013 (UTC)

Touché. Anyway, I overreacted; I would be just fine merging the two, but it leaves a big disconnect between the research topics and the basic topics, and I had seen several people on this page argue they were different.Brirush (talk) 03:49, 30 November 2013 (UTC)

Oh, well, that's always a headache, I agree. But if they were split by level of difficulty, I'm not even sure which article should be more difficult. Is there a reason that gentop should be about research topics and PST about basic ones, or vice versa?

I don't have a good answer to the "level of difficulty" conundrum, but I doubt that it makes sense to split along those lines between these two titles. My current feeling, subject to being convinced otherwise, is that there should be a central article at one of the titles, and it should be mostly "basic", insofar as a subject that university math majors often first encounter in their third year can be "basic". Then articles about more specific topics can be linked to from that article, and they can be more advanced. --Trovatore (talk) 03:58, 30 November 2013 (UTC)

I think that that is very reasonable. You've given me some ideas. I can merge them if you like; I've written a mock-up on my user page: https://en.wiki.x.io/wiki/User:Brirush. I've just added a 'research areas' sectiin under all the basics. I'm going to bed, but I'll merge them tomorrow unless you do it first or someone objects.Brirush (talk) 04:09, 30 November 2013 (UTC)

Looks beautiful, at a glance. I'll have to find some time and really read it. Thanks for your efforts! --Trovatore (talk) 20:14, 30 November 2013 (UTC)

Wow, the second paragraph is gorgeous. I think you've maybe come up with a solution for a longstanding issue in the lead of the compact space article; you might want to bring it up on that article's talk page (or perhaps I will). I'm not sure about the third (one-sentence) paragraph though — what does it mean? I'm tempted to say just leave it out. --Trovatore (talk) 21:38, 30 November 2013 (UTC)

Thank you; I really appreciate it. The third sentence was an afterthought,and it's getting the axe. Brirush (talk) 02:20, 1 December 2013 (UTC)

Summary of compactness

I know that we've had trouble summarizing the notion of compactness in an imprecise way on this site, but the summary given here is wrong. It says:

Compact sets are those that can be covered by finitely many sets of arbitrarily small size.

But, given ε>0, I can cover the open interval (0,1) with finitely many open intervals of length ε - yet (0,1) is certainly not a compact space.  J.Gowers  21:27, 4 June 2014 (UTC) — Preceding unsigned comment added by J.Gowers (talk • contribs)

Oh, right, that is kind of the natural reading, isn't it? I was thinking of it as a restatement of the every-open-cover-has-a-finite-subcover version, with the arbitrary open cover defining the "arbitrarily small size". But that's probably not intuitive to anyone not already used to that definition. Rats. I was so impressed when I first saw it. --Trovatore (talk) 21:33, 4 June 2014 (UTC)

The bullet-point summary still reads: "Compact sets are those that can be covered by finitely many sets of arbitrarily small size". This is still wrong, because the restriction that those sets are of "arbitrarily small size" cannot apply to the most general topological spaces, since only metric spaces have any relevant concept of size.

Is the following better: "A compact set is one that, given any collection of open sets that can cover it, can also be covered by just finitely many of those open sets"? Whether we can rely on the reader's intuition about what the verb "cover" means, I just don't know. But we can't assume they'll know what the noun "cover" means in topology. Do we need to define this term before discussing compactness? yoyo (talk) 06:23, 26 November 2018 (UTC)

Foundations of topology

I see such subsection (11.6), with only the link "Main article: Topology"; and I do not see anything like that in "Topology" article. What is meant by "Foundations of topology"? And more generally: separate foundations for each branch of mathematics? Or "federal" foundations of the whole mathematics? Boris Tsirelson (talk) 06:31, 6 February 2015 (UTC)

When I rewrote the article, I used online lists of topics related to general topology. I wasn't sure what "foundations of topology" was either. Feel free to omit it.Brirush (talk) 12:32, 6 February 2015 (UTC)

Well, if so, I just did. Whoever knows whats'it, feel free to insert it. Boris Tsirelson (talk) 15:28, 6 February 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:General topology/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Expand history and scope; add material on applications and major theorems. Tompw 18:30, 5 October 2006 (UTC)

Last edited at 16:57, 14 April 2007 (UTC). Substituted at 02:08, 5 May 2016 (UTC)

Errors in the section "Defining topologies via continuous functions"

There are at least two mistakes in this section: http://math.stackexchange.com/questions/2101750/continuous-functions-define-a-topology . I don't think I'm competent to clean this up, and I'm not sure if I understand the intention of the author.--76.169.116.244 (talk) 23:29, 17 January 2017 (UTC)

Yes, thank you, I see the problem(s). Yes, the intention of the author is vague, since the class of all continuous functions into all topological spaces must satisfy some conditions, not specified here. (The same problem for the dual approach.) On the other hand, it is easy to see that this class determines uniquely the given topology. Indeed, given two topologies $\tau _{1},\tau _{2}$ on $S$, we may consider all continuous functions $S\to X_{k}$ for $X_{1}=(S,\tau _{1}),X_{2}=(S,\tau _{2})$ and check continuity of the identity maps $f_{1}:S\to X_{1},f_{2}:S\to X_{2}.$ Clearly, $f_{1}$ is continuous w.r.t. $\tau _{1},$ and $f_{2}$ is continuous w.r.t. $\tau _{2}.$ Now, if $\tau _{1},\tau _{2}$ have the same continuous functions, then also $f_{1}$ is continuous w.r.t. $\tau _{2},$ and $f_{2}$ is continuous w.r.t. $\tau _{1},$ which means $\tau _{1}=\tau _{2}.$ Boris Tsirelson (talk) 07:49, 18 January 2017 (UTC)

Apparently that section was copied from Continuous function to Point-set topology (diff) which was later merged into General topology (diff). I.e., Continuous function#Defining topologies via continuous functions still contains very much the same text. Furthermore, a variant of that text first appeared at Continuous function (topology) which had then been merged into Continuous function (diff). Anyway, any mistake that is fixed at General topology#Defining topologies via continuous functions probably needs to be fixed at Continuous function#Defining topologies via continuous functions, too.—Tea2min (talk) 09:35, 18 January 2017 (UTC)

Wow... Then, I'll just do the same fix there. Thanks. Boris Tsirelson (talk) 09:58, 18 January 2017 (UTC)

In fact, the author was User:Jakob.scholbach, Aug 2011. Boris Tsirelson (talk) 10:12, 18 January 2017 (UTC)

Proposed reference to algebraic general topology

I propose to add a reference to Algebraic_General_Topology._Volume_1 because it is a notable, authoritative work important in the context of general topology and relevant for the content of this article as a natural generalization of general topology.

Probably we should also create a page about algebraic general topology as a math branch (not just as about a book).

Huh?

--VictorPorton (talk) 22:40, 1 October 2019 (UTC)

As you can see on your own talk page, the article Algebraic_General_Topology._Volume_1 has already been deleted twice.—Anita5192 (talk) 01:04, 2 October 2019 (UTC)