G-delta subsets of locally compact sober spaces

A kind of space that has good properties and that I regularly bump into are the Gδ subspaces of locally compact sober spaces.  For example, every continuous valuation on such a space automatically extends to a measure on its Borel σ-algebra.  Or: every such space is consonant.  Or: projective limits of directed systems of locally finite continuous valuations (where the index set has a countable cofinal subfamily) on such spaces exist and are unique [1].

How should I call those spaces?  “Gδ subspace of a locally compact sober space” is longish…

I recently asked the question to Matthew de Brecht, because those spaces include his own quasi-Polish spaces, without the separability requirement.  His angle was rather to see whether that kind of space was already known under another name, and he had the intuition that that should be some notion of completeness.  Hence call them “X-complete”, for some unknown X.

Let me see.

  1. Every locally compact sober space is X-complete (sure).
  2. Every complete metric space, in its open ball topology, is X-complete.  This is because the space embeds in its space of formal balls, which is then a continuous dcpo [2].  That includes all Polish spaces.
  3. Every continuous complete quasi-metric space, in its d-Scott topology, is X-complete.  The reason is similar [3].  That includes all quasi-Polish spaces.
  4. Every T0 completely regular Čech-complete space is X-complete.  This is because, by Frolík’s Theorem (Exercise 7.6.22 in the book) those spaces are exactly those that embed as a (dense) Gδ subspace of a compact T2 space (its Stone-Čech compactification).

On the other hand:

  1. Every X-complete space is Choquet-complete.  This is because every locally compact sober space is Choquet-complete (Proposition 8.3.24 in the book; α even has a stationary winning strategy) and a Gδ subspace of a Choquet-complete space in which α has a stationary winning strategy is Choquet-complete.  (I cannot find the latter in the book, damn.  But that has an easy proof.  Let X be the intersection of a decreasing sequence Wn of opens in Y.  At the nth turn played in X, α looks at the open subset V of X that β has just played.  This is the intersection with X of an open subset V’ of Y.  Then α simulates the stationary winning strategy it has on Y to find a smaller open subset U of Y, and plays U ∩ Wn.  Note that this new strategy is not stationary, but it is Markov, in the sense that it only depends on what β has just played and n, only.  It is also true that Gδ subspace of a Choquet-complete space is Choquet-complete, without the need to rely on stationary strategies, but the argument is a bit more complicated.)

Matthew de Brecht thought that X-completeness sounded a lot like Čech-completeness, and that maybe we could replace the Stone-Čech compactification in the proof of item 4 by some non-Hausdorff “compactification”.  Proposition 18 of [4] seems to be exactly what we need… but there is a gap in the proof.

This leads to a whole set of questions:

  1. Is every X-complete space Čech-complete, in the weak sense that I am using in Exercise 7.6.21 of the book?  That notion implies Choquet-completeness.
  2. Is every Čech-complete space X-complete?
  3. Is every X-complete space also a Gδ subspace of a stably (locally?) compact space?  Embedding a space X into some space of filters of open subsets of X may be a good idea, following [4] (see also Exercise 9.3.10 in the book).
  4. Can we characterize X-complete spaces by some form of (Choquet, Banach-Mazur) game?

Update (March 20, 2019).  With M. de Brecht, X. Jia, and Z. Lyu, we have put everything we know about those spaces in a paper submitted to ISDT 2019.  We have decided to call these spaces LCS-complete—LCS stands for “locally compact sober”.

Question 8 has a negative answer, by the way: every subspace of stably locally compact space is coherent, and there are non-coherent LCS-complete spaces, for example any non-coherent continuous dcpo, such as the set of negative integers with two incomparable elements added below all of them.

  1. Jean Goubault-Larrecq.  Products and projective limits of continuous valuations on T0 spaces.  arXiv:1803.05259 [math.PR]
  2. Abbas Edalat and Reinhold Heckmann.  A computational model for metric spaces.  Theoretical Computer Science Vol. 193, 1998, pages 53-73.
  3. Jean Goubault-Larrecq and Kok Min Ng. A few notes on formal balls.  Logical Methods in Computer Science Vol. 13(4:18), 2017, pages. 1–34.
  4. Michael B. Smyth. Stable compactification I. Journal of the London Mathematical Society s2-45, Issue 2.  April 1992, pages 321-340.

— Jean Goubault-Larrecq (March 30th, 2018)jgl-2011