Continuous functions on limits of F-decomposable systems

We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the images of any two closed disjoint subsets is finite. We give a characterization of such systems in terms of a property of the continuous functions on their limit. When, moreover, the fibers of neighboring bonding mappings are metrizable, we call the limit of such a system an F_d-compact, a particular case of a Fedorchuk compact. The stated property allows us to obtain a locally uniformly rotund renorming on the space C(K), where K is an F_d-compact of countable spectral height.