Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

A. Eysen, A. Leiderman, V. Valov

TL;DR

This paper investigates when dimensional-like and related topological properties transfer from a metrizable space $X$ to a perfectly normal space $Y$ under uniformly continuous surjections between $C_p$-type function spaces, specifically $T: D_p(X)\to D_p(Y)$ with $D\in\{C,C^*\}$ and $D_p$ denoting the pointwise topology. It develops a general scheme based on Gul'ko's support method and the notion of inversely bounded maps to show that if $T$ is uniformly continuous and inversely bounded, and $X$ has a property $\\mathcal{P}$ satisfying a set of closure conditions, then $Y$ also has $\\mathcal{P}$. This yields inheritance results for zero-, countable-, and strongly countable-dimensionality, as well as for scattered, strongly $\\sigma$-scattered, and $\\Delta_1$-spaces, extending and strengthening prior work. The paper also provides linear-operator analogues and a unified treatment across the four cases of $C_p$ vs $C_p^*$, contributing new insights into the interplay between the geometry of $C_p$-spaces and topological dimension-like properties. Overall, the results have implications for understanding how topological complexity is preserved under surjective maps between function spaces endowed with the pointwise topology.

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 $σ$-scattered space, or being a $Δ_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].

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

TL;DR

This paper investigates when dimensional-like and related topological properties transfer from a metrizable space to a perfectly normal space under uniformly continuous surjections between -type function spaces, specifically with and denoting the pointwise topology. It develops a general scheme based on Gul'ko's support method and the notion of inversely bounded maps to show that if is uniformly continuous and inversely bounded, and has a property satisfying a set of closure conditions, then also has . This yields inheritance results for zero-, countable-, and strongly countable-dimensionality, as well as for scattered, strongly -scattered, and -spaces, extending and strengthening prior work. The paper also provides linear-operator analogues and a unified treatment across the four cases of vs , contributing new insights into the interplay between the geometry of -spaces and topological dimension-like properties. Overall, the results have implications for understanding how topological complexity is preserved under surjective maps between function spaces endowed with the pointwise topology.

Abstract

Let be metrizable, be perfectly normal and suppose that there exists a uniformly continuous surjection (resp., ), where (resp., ) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on endowed with the pointwise convergence topology. We show that if additionally is an inversely bounded mapping and has some dimensional-like property , then so does . For example, this is true if is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties : of being a scattered, or a strongly -scattered space, or being a -space (see [17]). Our results strengthen and extend several results from [6], [13], [17].
Paper Structure (4 sections, 11 theorems, 3 equations)

This paper contains 4 sections, 11 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem-main']}
  3. Proof of Theorem \ref{['linear']}
  4. Scattered-like properties $\mathcal{P}$

Key Result

Theorem 1.2

Let $\mathcal{P}$ be the property of metrizable spaces such that: Then, for metrizable spaces $X$ and $Y$ such that $C_p(X)$ and $C_p(Y)$ are uniformly homeomorphic, $X\in \mathcal{P}$ if and only if $Y\in \mathcal{P}$.

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • Theorem 4.1
  • ...and 10 more