Talk:Compact space

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 Definition A toplogical space for which every covering with open sets has a finite subcovering. [d] [e]
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Compact set vs compact space

Don't you think this article should be rather a subsection in more general compact space? Wojciech Świderski 05:28, 12 July 2008 (CDT)

The terms compact set and compact space mean almost the same to me. Could you please explain the difference? -- Jitse Niesen 09:33, 12 July 2008 (CDT)
In general, a compact set is part of surrounding topological space that may not be compact - as closed and bounded subsets of R^n. Compact space is "compact in itself" - we don't think of it as of part of something greater. Compact manifold is a good example - if you don't consider it as embedded in anything else. See: [1] Wojciech Świderski 03:10, 13 July 2008 (CDT)
Okay, then we're using the same definitions. I was a bit surprised by your statement that "compact space" is more general than "compact set", but I guess it depends on how you look at it. Anyway, feel free to extend the discussion in the article. I do believe that "compact space" and "compact set" mean more or less the same (at least, the definitions are the same). Every compact set can be viewed as a compact space, if you forget about the space it's embedded in; every compact space is also a compact set in the space itself. So I think it's best to discuss both concepts in the same article. -- Jitse Niesen 16:19, 13 July 2008 (CDT)

I would really like to retitle this as "compact space". Compactness is a property of a topological space, and it seems odd that the latter concept isn't even mentioned in the introduction. Richard Pinch 19:05, 30 October 2008 (UTC)

Go ahead. -- Jitse Niesen 12:24, 31 October 2008 (UTC)

Properties

"The quotient topology on an image of a compact space is compact.
The image of a compact space under a continuous map to a Hausdorff space is compact."

No, the matter is simpler: The image of a compact space under a continuous map is compact. Boris Tsirelson 15:10, 25 May 2010 (UTC)

Compactness axioms

Why "compactum" redirects to "Compactness axioms" rather than here? And why the Compactness axioms page exists at all, separately from this page? Boris Tsirelson 15:53, 25 May 2010 (UTC)

I agree. Compactness axioms should be merged into Compact space.

Eric Toombs 19:16, 18 October 2010 (UTC)

Some things I don't understand

If a subcover B is a strict subset of a cover A, meaning B cannot equal A, then I don't see how a closed subset of R^n could be compact. A could have only one set, making B the empty set. This is why I think a subcover can't be a strict subset. If it can't, then in the definition of a subcover should be .

Also, an example of why a bounded open subset of Euclidean space is not compact would be helpful. First, though, which topology does the Heine-Borel theorem use on the subset of R^n? I am assuming it uses the subspace topology.
Eric Toombs 03:49, 19 October 2010 (UTC)

(a) Unfortunately, some authors interprete as strict inclusion, others as non-strict inclusion. It makes a problem both to Wikipedia and here. Even worse, on early (undergraduate) stage 'strict' is more typical, but later, on the graduate stage and in the math journals, 'non-strict' is more typical, and then the is not used. The latter interpretation is meant in this article.
(b) Of course, the subspace topology. It is the usual convention that, unless otherwise stated explicitly, R^n is endowed with its usual topology, and every subset of a topological space is endowed with the subspace topology. (Indeed, when you say '2+2=4" you probably do not specify, which binary operation is denoted by '+' this time!)
The problem is, to which extent such conventions should be included into every math article...
Boris Tsirelson 05:59, 19 October 2010 (UTC)