$\pi$-spaces and their open images

TL;DR

This paper introduces $ ext{-spaces},$ characterizes them via Souslin schemes, and explores their open images, establishing connections with Lusin $ ext{-bases}$ and second-countability.

Contribution

It defines $ ext{-spaces},$ characterizes their properties, and provides a main result describing their continuous open images in terms of topological bases.

Findings

Every space with a Lusin $ ext{-base}$ is a $ ext{-space}$.

Every second-countable $ ext{-space}$ has a Lusin $ ext{-base}.

Main result characterizes continuous open images of $ ext{-spaces}$.

Abstract

We study spaces that can be mapped onto the Baire space (i.e. the countable power of the countable discrete space) by a continuous quasi-open bijection. We give a characterization of such spaces in terms of Souslin schemes and call these spaces $\pi$-spaces. We show that every space that has a Lusin $\pi$-base is a $\pi$-space and that every second-countable $\pi$-space has a Lusin $\pi$-base. The main result of this paper is a characterization of continuous open images of $\pi$-space.