Krzysztof Ciesielski - Each Polish space cocompactly quasimetrizable

On April of 1998 Ralph Kopperman and Bob Flagg asked me whether every for Polish space there exists a countable collection C of closed subsets of X such that:

(1)  each subset of C with the finite intersection property has nonempty intersection,

(2)  for every open set T and x from T there exists a C in Csuch that C is a subset of T and x belongs to the interior of C, and

(3)  for every C from C and x from the complement of C there exists a finite subcollection G of C such that C is contained in the interior of the union U of G and x is still in the complement of U.

I was able to answer this question positively. In fact, the constructed family satisfies condition (3) with the singleton families G. This fact implies, in particular, that the following properties are equivalent for every topological space X.

(A)  X is a Polish space.

(B)  X is cocompactly quasimetrizable, that is, X is a Lindelöf space arising from a quasimetric whose dual yields a compact (not necessarily T₂) topology.

(C)  X has a bounded complete approximation (computational) model, that is (loosely), the points can be encoded and approximated by sets containing them in a computer program.