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

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 for the function space \(C_p(X,[0,1])\) to be Baire for a Tychonoff space X all separable closed subsets of which are \(C^*\)-embedded. In particular, this characterization holds for normal spaces and, hence, for metrizable spaces. Moreover, we established that the space \(C_p(X,[0,1])\) is Baire if and only if \(C_p(X,\mathbb {K})\) is Baire for a Peano continuum \(\mathbb {K}\).