Completeness type properties on \(C_{p}(X,Y)\) spaces
The authors present results related to characterizations of several kinds of pseudo-completeness and compactness properties in spaces of continuous functions of the form \(C_p(X,Y)\). In particular, they prove that if \(X\) is Tychonoff, and \(G\) is a separable metrizable topological group, for which \(C_p(X,G)\) is dense in \(G^X\), then \(C_p(X,G)\) is weakly \(\alpha\)-favorable if and only if \(X\) is \(u_G\)-discrete (i.e. every countable \(C\subset X\) is discrete and each continuous \(f:C\to G\) is continuously extendable over \(X\)). This result yields two generalizations of a theorem of \textit{V. V. Tkachuk} [Sov. Math., Dokl. 27, 681--684 (1983; Zbl 0539.54001); translation from Dokl. Akad. Nauk SSSR 270, 795--798 (1983)]: (1) Let \(G\) be a separable completely metrizable topological group and \(X\) a set. If \(H\) is a dense subgroup of \(G^X\) which is homeomorphic to \(G^Y\) for some set \(Y\), then \(H = G^X\) (e.g. when \(X\) is Tychonoff, and \(H = C_p(X,G)\)). (2) Let \(G\) be a realcompact Čech-complete topological group with countable pseudocharacter, and \(X\) be regular \(C^G_{<\omega}\)-discrete (i.e. whenever \(F\subset X\) is closed, \(n<\omega\), \((x_1,\dots,x_n)\in (X\setminus F)^n\), \((y_0,\dots,y_n)\in G^{n+1}\), there is a continuous \(f:X\to G\) with \(f(x) = y_0\) for every \(x \in F\) and \(f(x_i) = y_i\) for all \(1\leq i \leq n\)). Then, \(C_p(X,G)\cong G^\kappa\) if and only if \(X\) is a discrete space of cardinality \(\kappa\). Properties related to weak \(\alpha\)-favorability, in particular, weak pseudo-compactness of \(C_p(X,G)\) are studied when \(G\) is a topological group; several open problems are posed.
scientific article; zbMATH DE number 6684523
scientific article