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Hello how to show the following:

Let $(X,\tau)$ be a topological space then a single point is compact but not necessarily closed.

Thank you!

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HINT: Let $X$ be the Sierpiński space: $X=\{0,1\}$, and the open sets are $\varnothing,\{1\}$, and $X$. Is $\{1\}$ a compact set in $X$? Is it a closed set?

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Take $\Bbb{R}$ with the trivial topology. The set $\{0\}$ is compact but is not closed.

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The topological space $X$ is $T_1$ (i.e. for any two distinct points $x$, $y$, $x$ has an open neighbourhood that does not contain $y$) if and only if all single-point sets are closed.

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