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.

  • 1
    $\begingroup$ You should add the example as an answer and (after any necessary delay) accept it. $\endgroup$ – David Roberts Jun 13 '18 at 2:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.