How can you tell if a specific topological space is compact?

1. Mar 30, 2005

PBRMEASAP

eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.

2. Mar 30, 2005

matt grime

Yes, it would, that is why we have lots of theorems to help: the product of compact spaces is compact. Any closed subspace of a compact space is compact. Every compact metric space is sequentially compact. Usually, though I think it would be easier to show something wasn't compact by finding an explicit open subcover without refinement. Plus we often know a lot about the topological space other than simply that it is a topological space.

3. Mar 30, 2005

PBRMEASAP

Thanks, I'll look into those theorems you listed. Also, I wanted to make sure I have the notion of a cover straight. If a family of sets form a cover for a set A, then that means A is contained in the union of those sets, right? The reason I ask is that on the Wolfram Mathworld site, they say
In this case, we are talking about the whole topological space X, so is that why they say X is the union of a family of open sets? I just want to make sure that when talking about the compactness of a subset A of X, it is okay for A to be a proper subset of its cover, rather than equal to it.

4. Mar 30, 2005

matt grime

If A is a subset of a topological space, then it is almost certain that A must be a proper subset of the cover, otherwise A if it were equal to the union of the cover would be open, which is usually a way to be "not compact".

5. Mar 30, 2005

mruncleramos

Why don't people just go to the library and read books instead of trying to learn it from the internet? One cannot learn a subject by reading encyclopedia entries.

6. Mar 30, 2005

PBRMEASAP

matt:
Thanks for clearing that up!

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