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If we have a family of compact sets $\{K_\alpha\} $ for some metric space then what can we say about their intersection, is it compact? if not provide me a counterexample. I know this is true in $\mathbb{R}^k$.

Cornman
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Lucas
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$\bigcap_\alpha K_\alpha$ is closed, being the intersection of closed sets. Being a closed subset of a compact metric space (one of the $K_\alpha$), it is itself compact.

WoolierThanThou
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  • Ohh! got it. Thanks, I forgot about that – Lucas Aug 20 '20 at 14:02
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    Where the fact that we have a metric space is used for the last statement. Closed subsets of compact sets are compact in a metric space. In general it does not have to hold. A similar question was asked before. https://math.stackexchange.com/questions/229792/intersection-of-finite-number-of-compact-sets-is-compact – Cornman Aug 20 '20 at 14:02