Talk:Filters in topology

Too long

This article is too long, which makes it hard to read and edit. Any ideas how it can be split? --Erel Segal (talk) 06:59, 4 August 2021 (UTC)

Yes. Information that is purely set theoretic, which is most of the first part of the article, can be placed into its own article and then this article can direct readers there. For example, this set-theoretic information could be moved to Filter (set theory), which is currently a redirection page. I do not think that it should be moved into the article Filter (mathematics) because that article uses the order-theoretic definition of "filter" (i.e. the power set a filter in that article) so moving information from this article into Filter (mathematics) would require a major rewrite.Mgkrupa 22:24, 1 December 2021 (UTC)

Per the discussions here and at Talk:Filter (mathematics), I copied content from these two articles (mostly from Filters in topology) to create the article Filter (set theory). I will update these 2 articles appropriately later. That page had been a redirect to Filter (mathematics)#Filter on a set tagged with Template:R with possibilities. Mgkrupa 19:19, 11 May 2022 (UTC)

Umm, you should now delete the copied content from this article. We now have not one, but two articles, containing almost the same information, both of them far too long for comfortable reading. 67.198.37.16 (talk) 22:35, 30 April 2023 (UTC)

Incorrect definition

Correct definition of "Image of a filter" is here: https://proofwiki.org/wiki/Image_Filter_is_Filter The old definition was wrong/unrelated, now it is okey. — Preceding unsigned comment added by Georgydunaev (talk • contribs) 13:45, 31 August 2022 (UTC)

Convergent refinement

The article compact space states that a space is compact if every filter on it has a convergent refinement. The word "filter" links to this article. This article does not contain the word "refinement" in it. The article on compact spaces is high priority, so its disappointing that a basic claim in it cannot be followed up here.

The analogous article on Filter (set theory) does not contain the word "convergent" nor "refinement". Given that this and that article are almost identical cut-n-paste versions of each-other, its unclear why one can talk about convergence in topologies but not sets. Surely, there's a reason, but it is buried in the avalanche of text that it simply cannot be found.

The article Filter_(mathematics) does explain what a refinement is. It also talks about "convergence to a point". It is not clear where else the word "convergence" can be used. For example, the infinite intersection of all sets in a filter is a kind-of-ish convergent onto, well, the thing that the filter is filtering (a "closed set", usually).

Somehow, all this needs to be fixed up. 67.198.37.16 (talk) 00:59, 27 November 2023 (UTC)

About the definition of limit/convergence

@Mgkrupa In the Filters in topology#Limits and convergence section the definition of convergence of a filter (or more general family) is given both in terms of a limit being a point and in terms of a limit being a set. And Narici & Beckenstein is given as a source for this. This is misleading. Narici & Beckenstein only give the notion of a filter converging to a point, not to a set. And the same is true in other sources, like Willard, Engelking, Bourbaki. I think the section would be clearer if we rephrase the whole paragraph only in terms of convergence to a point. And then a sentence could be added at the end to mention that one can define a similar notion for convergence to a set. But that would have to be referenced in some reliable source from the literature. Of course you and I know it's trivial to define such a thing. But without such a reference, it would considered WP:OR and thus have no place here. And even if there is such a reference somewhere, that does not necessarily mean it must be mentioned here, if most of the other sources don't consider such a case, i.e., they consider a coherent and interesting theory can be developed without having to mention the notion of convergence of a filter to a set. Looking forward to your thoughts on this. PatrickR2 (talk) 03:27, 22 February 2024 (UTC)

Looking over it, convergence of/to sets should have remained in its own section (as it was originally) and I should have been more careful with the inline citations. So I removed it from the rest of the already very long article (remove anything that I missed). I'll add it back with proper citations when I have time later. In fact, there's enough written about convergence of sets/points to sets/points that it would make sense for the topic to have its own article, so I'll probably put it there instead of making this article longer. Mgkrupa 07:14, 24 February 2024 (UTC)

Sounds good. Thank you! PatrickR2 (talk) 02:52, 25 February 2024 (UTC)

Cluster point

It seems to me that the two sentences at the beginning of the section "cluster point" are saying the same exact thing. Or, am I missing something? Mennucc (talk) 10:50, 5 March 2024 (UTC)

$\wp$ as power set

There is interesting history behind the use of \wp to represent the power set - definitely a fun rabbithole. However, it totally threw me off and seems unnecessary to preserve that notational oddity. I would like to propose changing the symbol wherever it is used to the more standard \mathcal{P} (${\mathcal {P}}$). Tule-hog (talk) 01:23, 2 December 2024 (UTC)