Talk:Compact space

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Zundark, sorry that I screwed up the implications between the various compactness notions. --AxelBoldt

That's okay. I should have noticed it at the time. --Zundark, 2001 Dec 15

I'm new to Wikipedia, so I don't know if this is a good place to present a question. Prove or disprove: There exists a compact space X which can be covered by two (intersecting) open sets U and V so that no two compact sets K and F, with K a subset of U and F a subset of V, cover X. The example, if standard set theory can provide one, will be more complex than it seems at first glance. This is connected with properties of the compact-open topology. --Roman.

Contents 1 brackets 2 help 3 merge with Compact set proposal 4 Name Change Proposal 5 Mnemonical rule to remember Compact space 6 Compactness vs. completeness

[edit] brackets

Why is it important to nowiki the half-open interval, but leave the closed interval as is??? Revolver 11:46, 9 Nov 2004 (UTC)

[edit] help

Can someone help me understand this:

"The modern general definition calls a topological space compact if...any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space."

Does [0, 1] fit the definition of a compact space simply because there exist no collections of open sets whose union is [0, 1]? At least I cannot think of any--if you can think of some, please help me. Or should the word "is" be replaced by "contains"?

There are collections of open sets whose union is [0,1]. For example, { [0,.6), (.5,1] }. Remember that the open sets of [0,1] are the sets of the form [0,1]∩U, where U is an open set of R. --Zundark 07:32, 12 May 2005 (UTC)

I'm confused about this one too: taking the link open set's under metric space (R is a metric and topological space), we see that: A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. But the '1' in (.5,1] break that statement, no ?

Update: Ok, so the key is relative topology. I think someone should point that out in compact space, cover and possibly open set. The way it is now (4th August, 2005), we can deduce that a subset of a topological space is compact if it has a finite open subcover of its open cover. Then if you look up (open) cover, you conclude that a cover (of a subset S of a topology X) is a union of *open sets* in S which gives (hence the name "cover") the subset. Now, 'open' might be taken (which would be incorrect) to be from the original topology (and not the subspace one).

[edit] merge with Compact set proposal

It seems that compact set and compact space are rather duplicate. Shouldn't we merge these articles, taking care that both "unbounded and closed" and "finite subcover" are well treated and keeping the relation between the different definitions clear? I'm willing to do the merge myself, but comments/suggestions/objections are welcome. --Lenthe 11:29, 27 July 2005 (UTC)

You are right. It would however be preferable if compact set were merged into compact space, in such a way as to preserve the elementary flavour. That is, to be friendly first to those who need the idea on the real line, rather than the generality of topological spaces. Charles Matthews 12:04, 27 July 2005 (UTC)

I've been meaning to do this merge for a long time, but have never gotten around to it. Actually, it looks like much of the material in compact set belongs in the article on the Heine-Borel theorem. The rest of it should go in compact space. -- Fropuff 13:46, 27 July 2005 (UTC)

Ok, I've moved the content of compact set (now a redirect) to compact space and Heine-Borel theorem and started polishing both articles a bit. Help on polishing is very welcome. --Lenthe 14:59, 3 August 2005 (UTC)

[edit] Name Change Proposal

It might be more appropriate to name this page "compactness", especially considering the 'history' section. It isn't a history of compact spaces, but rather of compactness. It seems a more intuitive name as well. Fell Collar 01:51, 2 March 2006 (UTC)

Yet "compactness" can refer to a large number of things (even in mathematics), while "compact space" relates more clearly to the topological property. --Lenthe 08:40, 2 March 2006 (UTC)

What other things are you thinking of? Fell Collar 18:11, 2 March 2006 (UTC)

At any rate, the article isn't just about compact spaces. It's also about compact subsets of metric spaces or Rⁿ. Moreover, while "compactness" may refer to a number a things in mathematics, it seems to me that this is by far the most common. To clear up any confusion, we could always add a disambiguation link at the top of the article. --Fell Collar 15:40, 9 March 2006 (UTC)

First of all, didn't we just finish merging compact set into this article? It almost sounds like you want to split again (which I don't support). Second of all, the definition of compact sets in Rⁿ or metric spaces is simply that they are compact spaces in their subspace topologies. So the name "compact space" covers everything. I do not support a name change. -lethe ^{talk +} 16:43, 9 March 2006 (UTC)

I didn't mean to suggest we split the article again; I was suggesting that we keep all the content here, but change the name to something more inclusive of perspectives outside of topology that would justify the section about history as well. Compactness is also important in analysis, but one rarely refers to "compact spaces" there, nor does one reference topology in the definition of compact (Wheeden and Zygmund's "Measure and Integral", for example, refers only to "compact sets" and gives the finite open covering definition). This is a valid perspective as well. Rather than pick one point of view over the other, I'm suggesting we simply name the article "compactness" and list both perspectives. This seems rational to me, and in keeping with Wikipedia's NPOV guidelines. --Fell Collar 19:08, 9 March 2006 (UTC)

The appropriate name for this page is compact space. Please don't rename it. If you want to start a new article on compactness which discusses the various forms of compactness (sequential, countable, locally, para-) and their histories that would be fine. You could move some of the material from this page there. -- Fropuff 05:08, 10 March 2006 (UTC)

I'd be okay with that, if other people are as well. I expect there would be a considerable amount of redundancy with this article, which might be a concern given the recent merge. What do other people think? --Fell Collar 19:58, 10 March 2006 (UTC)

The only reason I would think that a separate page was needed would be if this page were too long. This page is currently not very long, so I do not support such a split, though I would change my mind if someone showed up who wanted to write all the stuff about those other forms of compactness which would make this article too long. Are you that person? -lethe ^{talk +} 22:59, 10 March 2006 (UTC)

I'm not particularly interested in writing all that, and I don't want this to be any more controversial than it already has been, so I'm just going to leave it alone. --Fell Collar 03:55, 11 March 2006 (UTC)

[edit] Mnemonical rule to remember Compact space

This rule came from Russia in a form of a joke: Mathematician is talking to a pretty girl: - You are so compact... Girl fondly specified his answer: - Do you mean well-shaped and thin? - No. Closed and bounded! --Yuriy Lapitskiy 22:26, 9 March 2006 (UTC) hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

[edit] Compactness vs. completeness

(Copied from user talk:AxelBoldt:) Do you happen to know the proof (or where it can be found) of the statement

A metric space X is compact if and only if every metric space homeomorphic to X is complete.

that you added to the compact space entry in 2002? Slawekk 17:41, 26 May 2006 (UTC)

I'm not so sure anymore that the statement is correct and I have removed it for now. The one direction is clear: if X is compact, then every space homoeomorphic to X is compact, and every compact metric space is complete. For the other direction, I wanted to use the Stone–Čech compactification βX of X: if βX is metrizable, then the subspace X of βX is a metric space homeomorphic to X and is therefore complete by assumption, and a complete subspace of a compact metric space is itself compact. Problem is, I'm not sure whether βX is metrizable. So I'm missing the following statement:

• The Stone–Čech compactification of every complete metric space is metrizable.

AxelBoldt 21:16, 26 May 2006 (UTC)

I don't think the missing statement is true. X has a metrizable compactification iff X is 2nd countable and tychonoff [1]. Of course not every complete metric space is second countable. Slawekk 22:24, 26 May 2006 (UTC)

You are right, thanks a lot! AxelBoldt 23:54, 27 May 2006 (UTC)