Baire property of some function spaces

Abstract

A compact space X is called \(\pi\)-monolithic if for any surjective continuous mapping \(f \colon X \rightarrow K\) 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 \(C_p(X,\mathbb{R})\) is Baire if, and only if, \(C_p(X,L)\) is Baire for each Fréchet 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 \(C_p(T,\{0,1\})\) is not.