## metric spaces – Baby Rudin Theorem 2.34

So here is the theorem

Theorem 2.34 The compact subsets of a metric space are closed.

Test.

Suppose $$K subseteq X$$, $$K$$ compact. Leave $$p in K ^ c$$, $$q in K$$. Leave $$V_q, W_q$$ be neighborhoods of $$p$$ Y $$q$$ with radius smaller than $$frac 1 2 d (p, q)$$.

As $$K$$ It's compact, we have $$K subseteq W_ {q_1} cup cdots cup W_ {q_n} = W$$ for some $$q_1, …, q_n in K$$.

If V =$$V_ {q_1} cap cdots V_ {q_n}$$, $$V$$ it's a neighborhood of p that doesn't cross $$W$$, so $$V subseteq K ^ c$$ so $$p$$ it is an interior point of $$K ^ c$$. QED

I don't see how compactness is relevant here. We just need a finite set of q, so that its neighborhoods cover K. But this does not mean that K is compact: there can be an open roof without a finite sub-cover.
(Actually, isn't there always a finite cover? Just take an open set that includes the set you want to cover)

Could you clarify? Thank you!

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