Collections of Compact Sets and Functions Having Gδ-Graphs

Abstract

We continue the investigation began in [1] of the connection between the structure of a function $f$ defined on a subset of a space $X$ and the Borel complexity of the set $C(f) =$ {$C \in J(X): f| C$ is continous where $J(X)$ denotes the nonempty compact subsets of $X$ with the Hausdorff metric. Two hierarchies of functions with $G_{\delta}$-graphs are defined. We conjecture that they coincide