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--- math104-s22:notes:lecture_13 [2022/02/28 23:19]
pzhou
+++ math104-s22:notes:lecture_13 [2026/02/21 14:41] (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