Once again on topological spaces containing a dense completely metrizable subset

Abstract

We characterize the existence of a dense completely metrizable subset in a regular topological space X via the existence of residually defined continuous single-valued selections for certain set-valued mappings with values in X. We show that the property “X contains a dense completely metrizable subset” is preserved by continuous and demi-open single-valued mappings. We show also that the existence of residually defined continuous selections implies the Closed Graph and the Open Mapping theorems for completely metrizable topological vector spaces.