On uniformly continuous surjections between $C_p$-spaces over metrizable spaces

TL;DR

This paper investigates how uniformly continuous surjections between certain function spaces over metrizable spaces preserve dimensional and topological properties of the underlying spaces.

Contribution

It establishes that under specific conditions, properties like zero-dimensionality and countable-dimensionality are preserved from the domain to the codomain space.

Findings

Preservation of dimensional properties under uniformly continuous surjections

Extension of previous results to broader classes of properties

Conditions involving inverse boundedness are crucial for property transfer

Abstract

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $\mathcal P$, then so does $Y$. For example, this is true if $\mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties $\mathcal P$: of being a scattered, or a strongly $\sigma$-scattered space, or being a $\Delta_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].