A relation between spaces implied by their \(t\)-equivalence (Q719521)

Let \(X\), \(Y\) be completely regular Hausdorff spaces and \(C_p(X)\) denote the space of continuous real-valued functions on \(X\) with the topology of pointwise convergence. We say that \(X\), \(Y\) are \(t\)-equivalent in case \(C_p(X)\), \(C_p(Y)\) are homeomorphic. In the present paper the author continues his study of topological properties preserved under \(t\)-equivalence by modifying some of his earlier results [Commentat. Math. Univ. Carol. 43, No. 2, 335--342 (2002; Zbl 1090.54004)]. We say that \(X\) is \(\sigma\)-discrete in case it is the union of a countable family of discrete subspaces, and \(X\) is \(F_\sigma\)-discrete in case it is the union of a countable family of closed discrete subspaces. The author shows, among other things, that for \(t\)-equivalent spaces \(X\), \(Y\): (i) if \(X\) is \(F_\sigma\)-discrete, then \(Y\) is \(\sigma\)-discrete; (ii) if \(X\) is metrizable, then every finite power of \(Y\) is a countable union of paracompact Fréchet spaces. A number of open questions are posed.