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 *T*_{3} 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 *T*_{1}. 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

Xbe countably-based andT_{1}.Xhas an ω-continuous modelYif and only ifXis 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, *T*_{1}, 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 *T*_{1 }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].

[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.