On topological Kadec norms

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let W ⊂ S be two separable metric topologies on the same set X. We prove that every point in X has an S-neighbourhood basis consisting of sets that are W-closed if and only if there exists a function φ: X→ ℝ that is W-lower semi-continuous and such that S is the weakest topology on X that contains W 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. © Springer-Verlag 2005.