Talk:Exhaustion by compact sets

Why change the article title to "exhaustion by subsets"?

@TakuyaMurata Why did you change the title of the page from "Exhaustion by compact sets" to "Exhaustion by compact subsets"? In the context of a topological space, there is no possible ambiguity: if a space X is going to be exhausted by a bunch of something, that something is necessarily something inside X, namely a subset of X. There is no other interpretation. And "exhaustion by compact sets" is also slightly shorter, and in common use too. See for example https://books.google.com/books?id=ZQVGAAAAQBAJ&pg=PA110&dq=%22exhaustion+by+compact%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj9r_zwz5D7AhVRD1kFHe_lAmYQuwV6BAgGEAc#v=onepage&q=%22exhaustion%20by%20compact%22&f=false (which is one of the references in the article). PatrickR2 (talk) 23:24, 2 November 2022 (UTC)

I'll give it a few more days and then revert the title if nobody wants to comment. PatrickR2 (talk) 08:53, 6 November 2022 (UTC)

The move has been completed. PatrickR2 (talk) 19:08, 9 November 2022 (UTC)

@PatrickR2: Sorry for not responding to you (I was aware from Wikipedia). I moved it not because of ambiguity but because compact sets here are compact subsets of a particular space; so I thought the latter might be clearer. I do however not have a strong opinion on the matter. -- Taku (talk) 08:51, 18 November 2022 (UTC)

About TakuyaMurata's latest changes (Nov 2024)

@TakuyaMurata: You did a bunch of changes to this page, and I may have some disagreements about some of them (comparing with the version before Nov 13).

First the lead. The previous lead had the definition, one quick example, and mentioned variant terminology. You added a bunch of information that does not belong in the lead. In particular, details about how to construct an exhaustion by compact sets in some cases. That is valuable information, but should not be in the lead. The lead is supposed to be a brief overview of the topic, and maybe a few very high level sentences (i.e., without any technical details) mentioning why this notion is important. (The previous version did not have , and neither does the current version; it's something that could be added.) In any case, more technical results (like what you added) should not be in the lead.

A space admitting an exhaustion by compact sets is called exhaustible by compact sets. This belongs right next to the definition, as it is just the same thing, not a different notion. But you moved it to the section "Relation to other properties". That does not make sense.

The previous organization of the page was one section about Properties. I am not saying it was optimal, but there weren't many items, so nothing really wrong with that. You changed the organization to one section for "Application" and one section for "Relation to other properties". What's your idea here? Why a first section about "Application"? It seems that's also related to Properties. And why put "Aplication" before "Properties"? It seems that Application should go after? That organization is confusing.

The whole thing about these two sections could and should be organized differently and better, maybe merged and split differently for example, once we decide better what we want to cover.

The previous Properties section had: Every regular space exhaustible by compact sets is paracompact. You changed it to Every regular space that is a countable union of compact sets is paracompact. That is, you dropped "exhaustible by compact sets", so it's not even about this page anymore??? And is it even true anymore??? Why do that?

I have more comments about the rest of your changes, but let's first discuss those. Looking forward to your response. PatrickR2 (talk) 05:35, 20 November 2024 (UTC)

I think there are two matters: an organization and specific facts. I can agree maybe the lead now has more technical stuff than it should, and I’m in agreement with the principle that the lead shouldn’t be too technical. I will thus put them into some sections. I renamed the section called “properties” since the section wasn’t actually much about properties and more about having links to other concepts (like sigma-compact) and so I think it made sense to rename it. The application comes first since, in my opinion, how an exhaustion is used in practice is more important. I moved a sentence about “exhaustible by compact sets” since the lead doesn’t need to mention every terminology and I don’t think this is used much as far as I can tell. Finally, I changed the statement about a regular space since otherwise the statement is pointless; the space is already paracompact (and regular) if there is an exhaustion so saying it is paracompact makes no sense. By the way, the changed statement is correct and meaningful (since a regular (Hausdorff) Lindelöf space is paracompact.) My suspicion was that the statement seems unrelated but I wasn’t 100% sure so I left it. But perhaps it should just be removed.

The fundamental question to me is who are the main audiences of this article. My sense is that should be readers interested in differential geometry or analysis and not those in general topology. This is in particular why the properties section was unhelpful and thus I added the application section before that. —- Taku (talk) 07:12, 20 November 2024 (UTC)

I disagree about the audience. This has as much to do with general topology as anything else. But why this is a useful tool for analysis or differential geometry is something that could be mentioned in the lead. You are probably more qualified than me about adding something there. (Please keep it short for the lead!). Apart from that, I don't want to go into an edit war. But as you boldly went ahead with sweeping changes, I intend to do something similar, but piece by piece so we can discuss along the way, to get to what I think would be a better organization of the material. Regards. PatrickR2 (talk) 01:26, 21 November 2024 (UTC)

P.S. I guess a regular Lindelöf space might be relevant to this article since it gives an example of a paracompact space that doesn’t admit an exhaustion by compact subsets (since it’s not necessarily locally compact). (By regular, I always mean regular Hausdorff.) —- Taku (talk) 07:28, 20 November 2024 (UTC)

We should follow the convention of the rest of Wikipedia, namely regular space does not imply Hausdorff. With Hausdorff, the usual terminology is $T_{3}$. PatrickR2 (talk) 01:30, 21 November 2024 (UTC)

The terminology "exhaustible by compact sets" is a trivial convenience to refer to a space that admits an exhaustion by compact sets. A citation is not required for that. It's just the adjectival form of the same concept, but referring to the space itself. That should be placed just after the definition of exhaustion, as the two go together. It's the same thing as a metric, and a topological space admitting a compatible metric, which is called a metrizable space. Here for example are two references of the usage: https://www.google.com/books/edition/Lectures_on_Algebraic_Geometry_I/ytv4yI8ZN80C?hl=en&gbpv=1&dq=%22exhaustible+by+compact+sets%22&pg=PA70 and https://arxiv.org/pdf/1502.00178v1 (p.12). So I am putting this back.

And I will also remove what you added in the lead about relations to other properties. That does not belong there. Instead, you should add something about how exhaustions by compact sets are useful in differential geometry for example? No need to get technical in the lead.

More stuff: PatrickR2 (talk) 05:10, 21 November 2024 (UTC)

(1) I restored the use of the notation $A\subseteq B$ versus $A\subset B$, as the latter may be ambiguous. That notation is sometimes used for proper subsets. The former is never ambiguous. Also, there was no need to put the chain of compacts on a line by itself. PatrickR2 (talk) 05:12, 21 November 2024 (UTC)

(2) Regarding this part of the lead: Occasionally some authors drop the requirement that ... (which it seems you added in June 2022), what authors are you talking about? Most, if not all, sources I have looked at use exhaustion by compact sets with the present meaning and not in the sense of sigma-compact. What are these other sources? And if they are very minor ones, can this sentence be dropped? (wikipedia should focus about notable facts). PatrickR2 (talk) 05:18, 21 November 2024 (UTC)

(edit conflict) You said “This has as much to do with general topology as anything else.” But the question is: are the concept and term “exhaustion by compact sets” generally something that can be found in standard textbooks? For example, I looked at Willard, General Topology and I couldn’t find the term, although the construction may be used somewhere somehow. On the other hand, the construction can be found almost any differential geometry textbooks or analysis books. I also reinserted some simple facts in the lead. It is standard and helpful to included some basic facts like paracompact in the lead (since paracompact is often what one needs). I also changed the subsetneq to subset as the latter is more standard in analysis. P.S. I didn’t add “Occasionally some authors…”. —- Taku (talk) 05:32, 21 November 2024 (UTC)

Let's start with something we can maybe agree on. The sentence "Occasionally some authors" was added by you on June 1, 2022. (See https://en.wikipedia.org/w/index.php?title=Exhaustion_by_compact_sets&diff=1090938923&oldid=1078941092 for the list of changes added by you on that day). Do you really want to keep it, or can we get rid of it? PatrickR2 (talk) 05:39, 21 November 2024 (UTC)

I just meant to say I didn’t add “occasionally some authors”. In hindsight, my initial wording was misleading: by sometimes, I meant in some other places in mathematics. I intended to mention that there is a weak condition that omits the requirement a set is contained in the interior. Subsequent edits by others misinterpreted the intended meaning (but of course, my initial wording should have be clearer). I will clarify this. —- Taku (talk) 05:47, 21 November 2024 (UTC)

I don't know why you keep saying you did not add "occasionally some authors". Please look at the diff from before and after your changes from Jun 1, 2022. You will see that the whole sentence was added by you. PatrickR2 (talk) 07:08, 21 November 2024 (UTC)

The addition was “Sometimes the requirement that $K_{i}$ is in the interior of $K_{i+1}$ is dropped (and, in that case, the existence of an exhaustion by compact sets means the space is σ-compact space.)”. I didn’t add “Occasionally some authors”. Like I said the intended meaning was to mention a weaker condition and like I said sometimes was a wrong word choice there. I am not denying of addition of a sentence and I am denying that the sentence current in the article is what I intended to mean. If subsequent edits have changed the intended meaning, why should I still be responsible for that. And I have already admitted “sometimes” was a wrong word choice. —- Taku (talk) 07:33, 21 November 2024 (UTC)

Please accept my apologies. I made a mistake. PatrickR2 (talk) 19:51, 21 November 2024 (UTC)

Also, why did you remove the example Q does not admit an exhaustion? It’s usually a good idea to give a non-example in the lead. —- Taku (talk) 06:42, 21 November 2024 (UTC)

It's good to mention that Q does not admit an exhaustion. But not necessarily in the lead. This can be expanded on later, which I plan to do. PatrickR2 (talk) 07:06, 21 November 2024 (UTC)

The removal usually means either the sentence is incorrect or irrelevant. And it is common to mention an example and a counter example in the lead; see e.g., Lie algebra. —- Taku (talk) 07:33, 21 November 2024 (UTC)