Xiaodong Jia asked the following question: is there any dcpo that is core-compact but not locally compact in its Scott topology? If every dcpo were sober, the answer would be no, but precisely, there are non-sober dcpos, such as Johnstone’s space (Exercise 5.2.15 in the book).
He also asked the following: is every well-filtered core-compact space locally compact? Every sober core-compact space is locally compact, but well-filteredness is a weaker property (except in the presence of… local compactness). Is every well-filtered core-compact dcpo locally compact? (Update of 24 sep 2019, 20 oct 2019:) Yes, that is true, see this post.
Every complete lattice is well-filtered, but not necessarily sober in its Scott topology. Is every core-compact complete lattice locally compact? (Update of 5 feb 2019:) Yes, that much is true. Indeed, as Xiaodong Jia tells me, this follows from results that he mentioned in his PhD thesis: by Theorem 2.5.8 there, every core-compact join-complete dcpo (i.e., such that every non-empty subset has a sup) is sober, hence locally compact; it even follows that a core-compact complete lattice is stably compact (Corollary 4.1.3 of his PhD thesis; the result is due to Gierz and Hofmann in 1977).
Then replace ‘dcpo’ by ‘monotone convergence space’: is every core-compact monotone convergence space locally compact?
That makes plenty of open questions… currently, the space X of [2, Section 7] (see also Exercise V-5.25 in ) is the only concrete example of a core-compact, non-locally compact space that we know. I have talked about it in my post of February 2019, by the way.
- Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. Continuous Lattices and Domains. Number 93 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003.
- Karl H. Hofmann and Jimmie D. Lawson. The Spectral Theory of Distributive Continuous Lattices. Transactions of the American Mathematical Society 246 (Dec. 1978), pages 285- 310.