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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)
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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