Fully closed maps and LUR renormability

Abstract

We show that the space of continuous functions over a compact space X admits an equivalent pointwise-lowersemicontinuous locally uniformly rotund norm whenever X admits a fully closed map onto a compact Y such that C(Y) and the spaces of continuous functions over the fibers all admit such norms. A map is called fully closed if the intersection of the images of any two closed disjoint sets is finite. As a main corollary we obtain that C(X) is LUR renormable whenever X is a Fedorchuk compact of finite spectral height.