Polish spaces, computable approximations, and bitopological spaces

Answering a question of J. Lawson (formulated also earlier, in 1984, by Kamimura and Tang [Theoret. Comput. Sci. 34 (1984) 275–288]) we show that every Polish space admits a bounded complete computational model, as defined below. This results from our construction, in each Polish space 〈X,τ 〉, of a countable family C of non-empty closed subsets of X such that: (cp) each subset of C with the finite intersection property has non-empty intersection; (br) if x ∈ T and T ∈ τ then there exists C ∈ C such that x ∈ int(C) and C ⊂ T ; and (r*) for every C ∈ C and x ∈X \C there is a D ∈ C such that C ⊂ int(D) and x / ∈D. These conditions assure us that there is another compact topology τ∗ ⊂ τ on X such that the bitopological space 〈X,τ, τ∗〉, is pairwise regular. The existence of such a topology is also shown equivalent to admitting a bounded complete computational model.  2002 Elsevier Science B.V. All rights reserved. AMS classification: 06B35; 06F30; 54C25; 54D30; 54D80; 54E50; 54E55; 54H12