Fully closed mappings and LUR renormability

We show that the space of continuous functions on a compact space $X$ admits an equivalent pointwise-lower-semicontinuous locally uniformly rotund norm whenever $X$ admits a fully closed mapping $\pi $ onto a compactum $Y$ such that $C(Y)$ and the spaces $C(\pi ^{-1}(y))$, $y \in Y$, all admit such norms. A mapping between compact spaces is called fully closed if it is continuous, surjective, and the intersection of the images of any two closed disjoint sets is finite. As a main corollary we show that $C(X)$ is LUR renormable whenever $X$ is a Fedorchuk compact space of finite spectral height.