Ideal presentations and numberings of some classes of effective quasi-Polish spaces

TL;DR

This paper extends the theory of ideal presentations to effective quasi-Polish spaces, providing new characterizations, numberings, and complexity estimates for these classes of spaces.

Contribution

It introduces effective domain-characterizations, natural numberings, and complexity analyses for classes of effective quasi-Polish spaces.

Findings

Effective domain-characterization of quasi-Polish spaces

Natural numberings of classes of effective quasi-Polish spaces

Complexity estimates of homeomorphism relations

Abstract

The well known ideal presentations of countably based domains were recently extended to (effective) quasi-Polish spaces. Continuing these investigations, we explore some classes of effective quasi-Polish spaces. In particular, we prove an effective version of the domain-characterization of quasi-Polish spaces, describe effective extensions of quasi-Polish topologies, discover natural numberings of classes of effective quasi-Polish spaces, estimate the complexity of the (effective) homeomorphism relation and of some classes of spaces w.r.t. these numberings, and investigate degree spectra of continuous domains.