Lusin spaces as images of locally compact Polish spaces

TL;DR

This paper studies the structure of c-Lusin spaces, which are images of noncompact locally compact Polish spaces, showing they can be decomposed into locally compact Polish parts and sets of discontinuity points, with applications to compactification.

Contribution

It introduces a decomposition theorem for c-Lusin spaces into locally compact Polish components and sets of discontinuities, extending understanding of their topological structure.

Findings

c-Lusin spaces can be expressed as a union of a locally compact Polish space and a set of discontinuities.

The set of discontinuities is closed and can be further decomposed similarly.

Under certain conditions, c-Lusin spaces are homeomorphic to compact metric spaces.

Abstract

A Lusin space is a Hausdorff space being the image of a Polish space under a continuous bijection. Such spaces have multiple applications, in particular, as state spaces of various stochastic systems. In this work, we consider the spaces obtained as the images of a noncompact and locally compact Polish space $(X, \mathcal{T})$, which we call $c$-Lusin. The main result is the statement that a $c$-Lusin space $Y=f(X)$, can be written as $Z\cup Y_1$, where $Z$ is a locally compact Polish space whereas $Y_1$ is $c$-Lusin. At the same time, $Y_1$ is the set of the discontinuity points of $f^{-1}$ which is a closed subset of $Y$. Moreover, $Y_1$ is nowhere dense if (and only if) $Y$ is a Baire space. By the same arguments, $Y_1$ can also be decomposed as $Z_1 \cup Y_2$ with the properties as above. In the case where $f$ can be extended to a continuous map $f:X\cup \{\infty\} \to Y$, and thus $Y_1$ is a singleton, we construct a metric on $X$ such that the corresponding metric space is compact and homeomorphic to the $c$-Lusin space $(f(X), \mathcal{T}')$.