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Unformatted text preview: Y,d Y ) are both compact metric spaces. Prove that X × Y is compact, when given the product metric. Hint: You can prove this directly by looking at open subsets of X × Y , but it may be easier to the result of problem 5: a metric space is compact if and only if and only if every inﬁnite subset of it has a limit point. If you try it this way, start building a limit point one component at a time. If you work directly from the deﬁnition by open covers, look at “slices” in one coordinate. 1...
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters

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