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.