Let $X$ be a compact topological space and $K$ be a compact subset of $X$. I want to prove that $K$ is a closed subset of $X$, or not. I tried to show that the complement of $K$ in $X$, is open. But, I am not able prove that. I also tried to show that $K$ contains all its limit points. If $K$ is not closed in $X$, please give an example.
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5Welcome to Mathematics Stack Exchange. I think this is true for Hausdorff spaces, but not for infinite sets with the cofinite topology; cf. answers to this question – J. W. Tanner Dec 21 '20 at 03:27
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Thanks for the answer. – Sudhir Kumar Dec 21 '20 at 11:44
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An infinite set in the cofinite topology gives a $T_1$ counterexample. Each of its subsets is compact. – Tyrone Dec 21 '20 at 15:02
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For a counterexample, let $K$ be the Sierpinski space, whose underlying point set is $\{0,1\}$
and whose open sets are $ \{\varnothing ,\{1\},\{0,1\}\}$. Since $K$ is finite, $K$ is compact.
The subset $\{1\}$ is compact but not closed.
J. W. Tanner
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Much simpler example: In indiscrete topology all subsets are compact but non empty proper subsets are not closed. – Kavi Rama Murthy Dec 21 '20 at 05:07
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