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--- math104-s22:notes:lecture_13 [2022/02/28 15:19]
pzhou
+++ math104-s22:notes:lecture_13 [2022/03/02 21:52] (current)
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 Topologically, we cannot distinguish spaces that are homeomorphic.
-===== Compactness =====
-There are two notions of compactness, they turns 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

Lecture Notes