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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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.
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.