Just a question to see if I understand the definition of a compact set: Is the set {0,1,2,3,4,5,6,7}, or any (finite) set $\displaystyle A\subset \mathbb{Z}$, compact?
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It seems to me that if you have an open cover of A, then you you choose for each element x of A a single open set that covers x. This way you get a finite subcover.
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