A characterization of trans-separable spaces

Abstract

The paper shows that a uniform space $X$ is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions $C_{c}(X) $ equipped with the compact-open topology is metrizable. This extends earlier results of Pfister and Robertson and also applies to show that if $C_{c}(X) $ is angelic then $X$ is trans-separable. The precise relation among DCCC spaces and trans-separable spaces has been also determined.