Compact Sets: Need help understanding

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My professor proved the following:

If C subset of X is a compact and A subset of C is closed then A is compact.

Proof: Let U_alpha be an open cover of A.

A subset of X is closed implies that U_0 = X\A is open.

C is a subset of (U_0) U (U_alpha) and covers X.

In particular they cover C (possibly containing U_0).

Therefore, a finite subcover exists U_0, U_1,...,U_k of C and these also cover A.
Therefore, U_1,U_2,...U_k, cover A and hence, A is compact.

I can follow the proof, but don't really see the idea behind it all. I don't see why is it that you just can't get the finite sub cover from C and say it also covers A so its compact. I guess maybe an open cover of A might not be included in the collection of all open covers from C, so it won't be for ALL open covers there is a finite cover.

Furthermore, why do do we kick out the U_0 = X\A at the end? Is there a reason why we have to?

If someone could explain what I'm having problems with, or even go through the entire proof with explanations it would be even better.

Thanks!

EDIT:

Wait, is the reason why I have to get the X\A out is that I'm technically selecting ONE open cover when I add it in? And so I must show that X\A goes for sure, so I only keep the very general U_alpha?
 

HallsofIvy

Science Advisor
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No, if we have any open cover for A, then adding the open set, X\A, to it gives an open cover for C. Since C is compact, any open cover for it has a finite subcover. The reason you can "kick" X\A out at the end is that you are looking for a finite subcover of A and X\A contains no points of A. The reason you need to "kick" X\A out is that you do not know if it was in the original cover of A.
 
273
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Thanks a lot. I get it now, you've helped me before and I appreciate it. Thanks!
 

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