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--- math104-s22:notes:lecture_14 [2022/02/28 15:25]
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+++ math104-s22:notes:lecture_14 [2022/03/02 21:59] (current)
pzhou
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 ====== Lecture 14: Compactness ======
+There are two notions of compactness, they turn out to be equivalent for metric spaces.
+Let  $X$  be a metric space,  $K \In X$  a subset.
+  * sequential compactness: we say  $K$  is compact, if every sequence in  $K$  has a convergent subseq.
+  * compactness: any open cover of  $K$  admits a finite subcover.
+The two notions turns out are equivalent, see https://courses.wikinana.org/math104-f21/compactness
+We will follow Pugh to give a proof. See also Rudin Thm 2.41
 We will finish discussion about compactness. In particular, in  $\R^n$ , we have Heine-Borel theorem, namely, compact subsets are exactly those subsets which are closed and bounded. Note that, this is false for general metric space, e.g. closed and bounded subset in  $\Q$  are not compact.

Lecture Notes