Baire property of spaces of $[0,1]$-valued continuous functions

TL;DR

This paper characterizes when the space of continuous [0,1]-valued functions on a Tychonoff space is Baire, linking it to properties of the underlying space and its subsets, with implications for normal and metrizable spaces.

Contribution

It provides a new characterization of Baire property for function spaces based on the topological features of the underlying space, especially for spaces with separable closed subsets.

Findings

Characterization of Baire property for $C_p(X,[0,1])$ spaces.

Equivalence of Baire property between $C_p(X,[0,1])$ and $C_p(X,K)$ for Peano continuum $K$.

Applicable to normal and metrizable spaces.

Abstract

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,[0,1])$ denote the space of all continuous $[0,1]$-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper, we have obtained a characterization when the function space $C_p(X,[0,1])$ is Baire for a Tychonoff space $X$ all separable closed subsets of which are $C$-embedded. In particular, this characterization is true for normal spaces and, hence, for metrizable spaces. Moreover, we obtained that the space $C_p(X,[0,1])$ is Baire, if and only if, the space $Cp(X,K)$ is Baire for a Peano continuum $K$.