Graphical convergence of continuous functions

Abstract

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈fλ〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.