Some topologies on the spaces of USCO maps and densely continuous forms

Abstract

We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity U B . We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ℝ) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in [7].