On the Behaviors of Hyperspace Topologies for Subspaces

摘要

For a topological space X we denote by CL(X) the collection of all nonempty closed subsets of X.Suppose we have a map T which assigns in some coherent way to every topological space X some topology T(X) on CL(X).In this paper we study continuity and inverse continuity of the map iA,X:(CL(A),T(A))→(CL(X),T(X))defined by iA,X(F)=F whenever F∈ CL(A),for various assignment T;in particular,for locally finite topology,upper Kuratowski topology, and Attouch-Wets topology, etc.