A topological completeness concept with applications to the open mapping theorem and the separation of convex sets
A topological space is called (strictly) p-complete by the author if it admits a (strictly) p-complete web. These spaces have previously been introduced game-theoretically by \textit{H. White}, who called the p- complete spaces weakly \(\alpha\)-favourable. Every Čech complete space is strictly p-complete, hence p-complete; and every p-complete space is Baire. The interesting applications of p-completeness presented in the paper are based on Theorem 1.2: A continuous dense and nearly feebly open mapping (i.e., int \(\overline{f(V)}\neq \emptyset\) for open \(V\neq \emptyset)\) from a p-complete space E to a metrizable space F maps residual subsets of E onto residual subsets of F. A consequence of this is Theorem 1.4 about the openness of a continuous nearly feebly open bijection between metrizable spaces. The concept of p- completeness is then used to define (strictly) pseudo-complete convex subsets of locally convex vector spaces. For Fréchet spaces E it is shown that every CS-closed subset of E in the sense of \textit{G. Jameson} and in particular every convex \(G_{\delta}\)-subset of E is strictly pseudo-complete, and that every pseudo-complete convex subset of E is semi-closed in the sense of Jameson. In Section 6, strictly pseudo- complete convex subsets of Fréchet spaces are characterized game- theoretically. In Section 4, an open mapping theorem for multifunctions between Banach spaces having a strictly pseudo-complete convex graph is proved which generalizes open mapping theorems of \textit{Jameson} and \textit{S. Robinson}. In Section 5, Tuckey's classical separation theorem is generalized by: Let C,D be disjoint strictly pseudo-complete convex sets in a Banach space E. Suppose that C is bounded and C-D is not nowhere dense. Then C,D can be separated by a closed hyperplane.
scientific article; zbMATH DE number 4194669
scientific article