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--- math104-s22:notes:lecture_15 [2022/03/08 06:45]
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+++ math104-s22:notes:lecture_15 [2026/02/21 14:41] (current)
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+. Compactness is an absolute (or intrinsic) property of a metric space. If $X$ is a metric space, and $K \In X$ is a subset, when we say $K$ is compact, we mean $K$ as a 'stand-alone' metric space (totally forgetting about $X$, but only using the distance function inherited from $X$) is compact.
+. We proved last time: If $K \In X$ is (open cover) compact, then $K$ is closed and bounded. (Discussion: if you replace open cover compact by sequential compactness, can you prove the two conclusions directly (without using the equivalence of the two definitions)?)
+. We know that $[0,1]$ is sequentially compact (by Heine-Borel theorem), can you show that $[0,1]^2$ is sequentially compact? (Hint: given a sequence $(p_n)$ in $[0,1]^2$, first show that you can pass to subsequence to make the sequence of the first coordinate converge, then ...)

Lecture Notes