Baire property of some function spaces

TL;DR

This paper investigates the Baire property of function spaces $C_p(X,Y)$, establishing conditions under which these spaces are Baire based on the properties of the domain and codomain spaces, with new constructions and equivalences.

Contribution

It characterizes when $C_p(X,Y)$ is Baire for various classes of spaces, introducing new equivalences and a specific example space with unique Baire properties.

Findings

$C_p(X, ext{{0,1}})}$ is Baire iff $C_p(X,K)$ is Baire for all $ ext{π}$-monolithic compact $K$.

$C_p(X)$ is Baire iff $C_p(X,L)$ is Baire for all Frechet spaces $L$.

Constructed a space $T$ where $C_p(T,M)$'s Baire property depends on $M$ being a Peano continuum.

Abstract

A compact space $X$ is called $\pi$-monolithic if for any surjective continuous mapping $f:X\rightarrow K$ where $K$ is a metrizable compact space there exists a metrizable compact space $T\subseteq X$ such that $f(T)=K$. A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,Y)$ denote the space of all continuous $Y$- valued functions $C(X,Y)$ on a Tychonoff space $X$ with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space $X$ the space $C_p(X,\{0,1\})$ is Baire if, and only if, $C_p(X,K)$ is Baire for every $\pi$-monolithic compact space $K$. For a Tychonoff space $X$ the space $C_p(X)$ is Baire if, and only if, $C_p(X,L)$ is Baire for each Frechet space $L$. We construct a totally disconnected Tychonoff space $T$ such that $C_p(T,M)$ is Baire for a separable metric space $M$ if, and only if, $M$ is a Peano continuum. Moreover, $C_p(T,[0,1])$ is Baire but $Cp(T,\{0,1\})$ is not.