How can I show this property of compact sets?

Question

How can I show that, if a subset A of R^n is compact, then every open cover of A contains a finite subcover?

Thank you

Answer 1

This is the DEFINITION of a compact set. You can't prove this, because this just the definition. It's like asking someone to prove a positive number is greater than 0.

There are some conditions equivalent to compactness, but the definition, valid in every topological space, is just the one you gave. Maybe you were asked to prove something else.

For example, if A is a subset of a metric space, then the following statements are equivalent

1- A is compact
2- Every infinite subset of A has limit points
3 -Every sequence in A contains a subsequence that converges to an element of A
4 -A is complete and totally bounded.

In Euclidean metric spaces, like R^n and the set of the complexes, 4 i equivalente to saying a set is compact if, and only if, it's closed and bounded. This is known as Heine Borel theorem.

In every topological space, even if not metric, 2 is still valid.

But, by DEFINITION, a set in a topological space is compact if every open cover of A contains a finite subcover.

Answer 2

Sorry, much to my chagrin, I can't answer it. But I had to thank you for posing a question that reminded me I really did do a maths degree [mmmmnph] years ago.

I might have to dig out some of my notes (or perhaps compact sets was A level). My wife's not going to like this...


Edit: Does this Wikipedia page help? http://en.wikipedia.org/wiki/Compact_set
It mentions a version of your question in relation to compact topological spaces and refers to the Heine-Borel theorem which, if the wiki entry is correct, proves exactly what you want.

Edit 2: Hmm, having read that wiki page and the one on covers/open covers/subcovers, it makes a little more sense, but not much. Still, I would be tempted to consider proof by contradiction (assume there exists an open cover of A which contain no finite subcovers...).

This question is going to give me some weird dreams tonight. Bless you for that. :-)

By the way, is no-one else going to answer this question? ;-)