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

[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.

 

[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.

 

[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

A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X.

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.

 

[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".

 

[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.

 

[PBRMEASAP]

matt:
Thanks for clearing that up!

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.

Who is you? Where you comin' from?