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This may be a stupid question, but I just confused myself on compactness. For some reason I can't convince myself that ANY subset of a compact set isn't compact in general; just closed subsets. Suppose K is a compact set and [tex]F \subset K[/tex]. Then if [tex](V_{\alpha})[/tex] is an open cover of K, [tex]K \subset \cup_1^n V_{\alpha}[/tex] for some n. But since [tex]F \subset K[/tex], doesn't that mean that [tex]F \subset \cup_1^n V_{\alpha}[/tex], which means that F...oh I just answered my own question. Ha. I guess it helps to write things out.

But just to make sure I understand what went wrong in the argument: The argument doesn't prove anything because it doesn't show that EVERY open cover of F contains a finite subcover. It just shows that every open cover of F that is also an open cover of K has a finite subcover, which is obvious.

But just to make sure I understand what went wrong in the argument: The argument doesn't prove anything because it doesn't show that EVERY open cover of F contains a finite subcover. It just shows that every open cover of F that is also an open cover of K has a finite subcover, which is obvious.

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