Some Hierarchies of QCB_0-Spaces

TL;DR

This paper introduces and analyzes hierarchies of topological spaces based on classical Borel and Luzin set hierarchies, focusing on their non-collapse properties and interrelations.

Contribution

It defines two classes of hierarchies for topological spaces, one for countably based spaces and another for more general spaces, and studies their structural properties.

Findings

Hierarchies do not collapse under certain conditions

Relationships between different hierarchy classes are established

Insights into the structure of QCB_0-spaces and their representations

Abstract

We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain P\omega, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.