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It is know that every compact subspace of Hausdorff space is closed and every closed set is compact.

So I have a question as folows: is there any compact non-Hausdorff space $X$ such that every open subspace of $X$ is compact?

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    $\begingroup$ It's not that every closed set is compact. In $\Bbb R^n$, a set must be closed and bounded in order to be compact. $\endgroup$ – Tim Raczkowski Mar 3 '15 at 16:25
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is there any compact non-Hausdorff space X such that every open subspace of X is compact?

Let $X$ be your favorite set with the trivial topology. Every subset of $X$ is compact, including the open ones.

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Any set endowed with the cofinite topology is compact, and all of its subspaces are compact.

Another example is $k^n$ with the Zariski topology, where $k$ is any field.

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