Tag: 234

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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