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We know from lemma 1.2.5 in part C of Sketches of an Elephant (by Johnstone) that both open and closed subspaces of a sober space are again sober. This raises the following question.

Question: Is a compact subspace of a sober space again a sober space?

Edit: Someone just told me a counterexample. Take the natural numbers with the upset topology and one point added on top. Then this is sober, but the subspace of natural numbers is a compact non-sober subspace.

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    $\begingroup$ You should add the example as an answer and (after any necessary delay) accept it. $\endgroup$ – David Roberts Jun 13 '18 at 2:52

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