The regular topology on \(C(X,Y)\) (Q2330047)

Let \(C_{r}(X,Y)\) denote the space of all continuous functions from a topological space \(X\) to a topological space \(Y\), equipped with the regular topology. In Section 2 the authors study the metrizability and countability properties of \(C_{r}(X,Y)\), where \(X\) is a Tychonoff space and \(Y\) is a metric space containing a nontrivial path. It is shown that metrizability of \(C_{r}(X,Y)\) is equivalent to many other weaker topological properties. In Section 3 the authors study various completeness properties of \(C_{r}(X,Y)\). Amongst others they prove that if \(Y\) is a complete metric space, then \(C_{r}(X,Y)\) is uniformly complete (Theorem 3.1) and \(C_{r}(X,Y)\) is a Baire space (Theorem 3.2).