A FAMILY OF GRAPH TOPOLOGIES FOR FUNCTION SPACES AND THEIR ASSOCIATED WEAK GRAPH TOPOLOGIES

Abstract

Let X, Y be topological spaces. We denote by Y^x , C(X, Y) and G(X, Y) the set of all functions from X to Y, the set of all continuous functions and of all functions with closed graphs respectively. Now as is well-known, identifying a function f with its graph Γfwe can transfer a (known) hyperspace topology for the product space X x Y to the function space Y^x (or to a subspace of Y^x ). A function space topology which is generated in this way we will call a graph topology. For a broad and important family of such graph topologies for each member of this family we can define an associated weaker graph topology. This family will be characterized, and we find general conditions implying the joint continuity and the validity of the HausdorfF separation axiom for the members of this family and their associated weak graph topologies. Applying these general assertions we get back corresponding results for the weak Fell topology and the Fell topology respectively and we get new results for graph topologies generated by connected sets.