Computability on Quasi-Polish Spaces

Abstract

We investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several characterizations. We show that some natural effectivizations of quasi-metric spaces are strictly stronger.Proposition 6.1. The space [α, 1] < is an effective quasi-Polish space iff α is left-c.e. relative to the halting set.

Effective Wadge Hierarchy in Computable Quasi-Polish Spaces

“…As shown in [15,3,4], CQP-spaces do satisfy effective versions of several important properties of quasi-Polish spaces. E.g.…”

“…Quasi-Polish spaces [2] have several characterisations. Effectivizing one of them we obtain the following notion identified implicitly in [15] and explicitly in [3,4]. Definition 1.…”

“…[11]) but, as also in the classical case, they behave well only for spaces of special kinds. Recently, a convincing version of a computable quasi-Polish space (CQP-space for short) was suggested in [3,4].…”

A syntactic approach to Borel functions: Some extensions of Louveau's theorem

“…One can also show that Theorem 5.1 holds for computable quasi-Polish spaces. Here, roughly speaking, a computable quasi-Polish space is a represented second countable T 0 space which is the image of a computable open surjection from ω ω (which almost corresponds to the effective version of Fact 4.2; see also [6,14]).…”

“…By effectivizing the arguments in Examples 5.5 and 5.6 to calculate the lightface complexity, we observe that Lemma 5.14 implies Theorems 5.2 and 5.3, which also deduces Theorem 5.1 for X = ω ω . To prove Theorem 5.1 for computable Polish X, first note that if X is computable Polish (indeed, if X is computable quasi-Polish; see [6,14]), then there exists a total computable open surjection δ : ω ω → X; see also Fact 4.2. Then for a function f :…”