A topological structure on a set of continuous functions with various domains (Q1972544)

A new topology on a set of continuous functions with various domains is introduced. Let \(U\) be an open subset of \(\mathbb{R}\times M\), where \(M\) is a metric separable locally compact space. By \(C_s(U)\), \(C^+_s(U)\) and \(C_s^-(U)\) are denoted the sets of all continuous functions of the form \(z:[t_1,t_2] \to M\), \(z:[t_0,t)\to M\), and \(z:(t,t_0]\to M\), respectively, whose graphs are closed in \(U\); \(t\) may be infinite. By \(C_s^\pm(U)\) we denote the set of all continuous functions \(z:(t_1,t_2)\to M\) whose graphs are closed in \(U\). Topologies on spaces \(C_s(U)\), \(C_s^+(U)\), \(C_s^-(U)\) and \(C_s^\pm(U)\) were studied in the literature and they are useful in the theory of ordinary differential equations. Define \(F(U)\) to be the union of \(C_s(U)\), \(C_s^+(U)\), \(C_s^-(U)\) and \(C_s^\pm(U)\). In the paper, a topology on \(F(U)\) is introduced such that the corresponding convergence is a natural generalization of uniform convergence for functions defined on a closed interval. Topologies induced by the topology of \(F(U)\) on \(C_s(U)\), \(C_s^+(U)\), \(C_s^-(U)\) and \(C_s^\pm(U)\) coincide with their known topologies. Metrizability properties of \(F(U)\) and its subsets are studied. For example, it is proved that \(C_s(U)\cup C_s^+(U)\) and \(C_s(U)\cup C_s^-(U)\) are metrizable and a subset \(X\) of \(F(U)\) is metrizable if and only if \(X\) is regular.