On topological Kadec norms

Abstract

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let Open image in new window be two separable metric topologies on the same set X. We prove that every point in X has an Open image in new window -neighbourhood basis consisting of sets that are Open image in new window -closed if and only if there exists a function φ: X→ℝ that is Open image in new window -lower semi-continuous and such that Open image in new window is the weakest topology on X that contains Open image in new window and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.