A model of a space X is a dcpo Y whose subspace of maximal elements is isomorphic to X. Of particular importance are those spaces that have ω-continuous models. Martin [3], and Mummert and Stephan [1] came very close to characterize them exactly.
Let’s review the notion. This is covered in Section 7.7.2 of the book. The prime example of an ω-continuous model is IR, the dcpo of all non-empty closed intervals of the real line R. We order the elements of IR by: [a, b] is below [c, d] if and only if [c, d] is included in [a, b]. Not the other way around! Higher in the model means a better approximation to a (possibly unknown) real value, hence a smaller interval. Maximal elements are one-point intervals [a, a], which we equate with the real number a. So, for example, a chain of better and better approximations to a real number may be [-2, 9], [1, 4], [2.9, 3.8], [3.1, 3.2], [3.14, 3.15], [3.141, 3.142], etc. (Can you say pi?) Models have been used for giving domain-theoretic foundations to computation over the reals, following this intuition, for one: see papers by Edalat, and by Escardò, in particular.
If a space Y has a ω-continuous model X, that gives you countable chains of approximations for any element in X, with good mathematical properties. A natural question, first attacked by Lawson [2], is: which spaces Y have ω-continuous models X at all?
Lawson showed that, if you require the model Y to occur as a subspace of X independently of the fact that X comes with its Scott or Lawson topology, then Y must be Polish; and that every Polish space Y has an ω-continuous model X.
Martin improved this [3] and showed that the T3 spaces that have ω-continuous models are exactly the Polish spaces. In particular, there is no hope of find a non-trivial theory of ω-continuous models for, say, analytic (Souslin) spaces.
The argument behind this is given in Theorem 7.7.23 of the book. The key ingredient found by Martin is that:
Any space that has an ω-continuous model must be Choquet-complete.
In fact, countability is irrelevant in this remark, and every space with a continuous model (not necessarily ω-continuous) is Choquet-complete. One can say even more: every space with a continuous model is convergence Choquet-complete, see Proposition 7.7.19 and Exercise 7.7.20 in the book.
Conversely, any space X with an ω-continuous model Y must be countably-based. Using Norberg’s Lemma (Lemma 7.7.13 in the book), indeed, X must be countably-based in its Scott topology, hence also its subspace Y.
Also, any space X with an ω-continuous model Y must be T1. This is because any two distinct maximal elements of Y must be incomparable.
Mummert and Stephan [1, Theorem 6.1 and Corollary 6.3] showed that:
Let X be countably-based and T1. X has an ω-continuous model Y if and only if X is Choquet-complete.
Moreover, we can even take Y to be ω-algebraic. The latter is relatively easy: if Y is an an ω-continuous model of X, then its ideal completion will be an an ω-algebraic model of X. The result quoted above is more intricate, and Mummert and Stephan obtain it through a study of so-called countably-based MF spaces, which, as they show, are exactly the countably-based, T1, Choquet-complete spaces, up to homeomorphism (their Theorem 5.3). An MF space is defined as the set of maximal filters of a poset, with the hull kernel topology. See [1] for more details.
This is rather amazing. This says that, provided we work with countably-based and T1 spaces, Choquet-completeness and having an ω-continuous model are the same thing.
Thanks to Matthew de Brecht for giving me a pointer to the paper [1].
— Jean Goubault-Larrecq (October 12th, 2012)
[1] Carl Mummert and Frank Stephan. Topological aspects of poset spaces. Michigan Mathematical Journal, 59, 2010, 3-24.
[2] Jimmie D. Lawson. Spaces of maximal points. Mathematical Structures in Computer Science, 7, 1997, 543-555.
[3] Keye Martin. The regular spaces with countably based models. Theoretical Computer Science, 305, vol. 1-3, 2003, 299-310.