Polish spaces, computable approximations, and bitopological spaces
The authors solve an open problem by showing that every Polish space can be the maximal point space of some bounded complete continuous domain with a countable base (bounded complete continuous domains are sometimes called continuous Scott domains). The bounded complete domain is called a computational model or domain environment for the given space. The proof depends on an intricate construction valid in any Polish space of a countable family \({\mathcal C}\) of non-empty closed sets that (i) forms a closed subbase for a compact topology \(\tau^*\), (ii) is a basis of (closed) neighborhoods for the original topology \(\tau\), and (iii) given \(C\in{\mathcal C}\) and \(x\notin C\), there exists \(D\in{\mathcal C}\) such that \(C\subseteq \text{int}(D)\) and \(x\notin D\). The authors show that the bitopological space \((X,\tau, \tau^*)\) is pairwise regular and then they use the existence of such a topology \(\tau^*\) to construct the bounded complete domain environment for the space.
scientific article; zbMATH DE number 1758562