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I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite subcover. To prove a set is compact in any arbitrary topological space, I know they must show that for every open cover there's a finite subcover, the problem is that I can't see intuitively how one could show this for every cover. Also when trying to disprove compactness the books I've read start presenting strange covers that I would have never thought about. I think my real problem is that I didn't get yet the intuition on compactness.

If the space is subset of $R^n$, then I can easily verify the compactness by using Heine Borel theorem,which is rather more geometrically motivated.

but can i give some geometrical arguments to answer compactness of set in arbitrary topological space.if such method exists, please give some references to show how to understand the process of proving (and disproving) compactness?

Martin Sleziak
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Abhishek Shrivastava
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  • In a metric space a set is compact if it is closed and bounded. Since $\mathbb{R}^n$ is metric with the Euclidean topology this is the most intuitive and geometric interpretation (for me). I think that more than a reference it can be useful to try to understand compactness for other topologies. Have you tried to see which sets are compact in the line $\mathbb{R}$ with the Zariski topology? Or in the plane $\mathbb{R}$ with the same topology? Or with other topologies (e.g. Sorgenfrey)? – User3773 Jan 05 '16 at 11:49
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    Maybe this post can be helpful for you: http://math.stackexchange.com/questions/485822/why-is-compactness-so-important?rq=1 – User3773 Jan 05 '16 at 11:52
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    @Cla Not "closed and bounded", but "complete and totally bounded". – Daniel Fischer Jan 05 '16 at 12:17
  • @Daniel, of course, sorry. I was thinking to $\mathbb{R}$ but for generic metric spaces I need something more. – User3773 Jan 05 '16 at 13:28

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