Separation functions and mild topologies

Abstract

Abstract Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 ( M ; N ) {C}^{0}\left(M;\hspace{0.33em}N) of continuous maps f : M → N f:M\to N . We review two classical topologies, the “strong” and the “weak” topology. We propose a definition of “mild topology” that is coarser than the “strong” and finer than the “weak” topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M → N f:M\to N , and affine actions when N = R n N={{\mathbb{R}}}^{n} . To define the “mild topology” we use “separation functions;” these “separation functions” are somewhat similar to the usual “distance function d ( x , y ) d\left(x,y) ” in metric spaces ( M , d ) \left(M,d) , but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define “set separation functions” that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C 0 ( M ; N ) {C}^{0}\left(M;\hspace{0.33em}N) . Other examples are the Sorgenfrey line and the topology of topological manifolds.