Wijsman topology on function spaces.

Let C(X,Y) be the space of continuous functions from a metric space ( X,d) to a metric space ( Y, e). C(X, Y) can be thought as subset of the hyperspace CL(X×Y) of closed and nonempty subsets of X×Y by identifying each element of C(X,Y) with its graph. We consider C(X,Y) with the topology inherited from the Wijsman topology induced on CL(X×Y) by the box metric of d and e. We study the relationships between the Wijsman topology and the compact-open topology on C(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology on C(X,Y) is weaker than the Wijsman topology are given (If Y is totally bounded, then for every metric space X the compactopen topology on C(X,Y) is weaker than the Wijsman topology and the same is true for X locally connected and Y rim-totally bounded). We prove that a metric space X is boundedly compact iff the Wijsman topology on C( X, ℝ) is weaker than the compact-open topology. We show that if X is a σ-compact complete metric space and Y a compact metric space, then the Wijsman topology on C(X,Y) is Polish. [ABSTRACT FROM AUTHOR]