Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts

Abstract

We show that C(X) admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided X is Fedorchuk compact of spectral height 3. In other words X admits a fully closed map f onto a metric compact Y such that f −1 (y) is metrizable for all y ∈ Y . A continuous map of compacts f : X → Y is said to be fully closed if for any disjoint closed subsets A, B ⊂ X the intersection f (A) ∩ f (B) is finite. For instance the projection of the lexicographic square onto the first factor is fully closed and all its fibers are homeomorphic to the closed interval.