$K\subseteq X$ compact, C closed, can I say something about $K \cap C$?

Question

If C is closed then C is compact as well, but the intersection of two compact set is not compact in general. It would be nice to have this intersection to be compact though, any idea if this is true?

If $C$ is closed it ($C$) need not be compact. But $C \cap K$ is. — Henno Brandsma, Jan 06 '18 at 17:53

Henno Brandsma · Accepted Answer · 2018-01-06T18:43:31.710

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$K \cap C$ is closed in $K$, by the definition of subspace topology, so compact. A closed subset of a compact space is always compact, no extra assumptions needed. $K \cap C$ need not be closed if $K$ is not closed.

edited Jan 06 '18 at 18:43

answered Jan 06 '18 at 17:37

Henno Brandsma

242,131

Why K $\cap$ C is closed in K though? – qcc101 Jan 06 '18 at 18:42
@LudovicoBessi subspace topology – Henno Brandsma Jan 06 '18 at 18:43
@Cornman $C$ is closed in $K$ makes no sense. It’s not even a subset of $K$. – Henno Brandsma Jan 06 '18 at 18:59
@LudovicoBessi $K$ need not be closed in $X$. – Henno Brandsma Jan 06 '18 at 19:05

$K\subseteq X$ compact, C closed, can I say something about $K \cap C$?

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