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:  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)
- "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)
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)