Irreducible representations of normal spaces

Abstract:

We define the notion of irreducible polyhedral representation of a normal space making use of approximate inverse systems. This generalizes the concept of irreducible polyhedral expansions introduced in 1937 by Freudenthal for metric compacta and generalized to uniform spaces by Isbell in 1961. We show that every normal space $X$ has an irreducible polyhedral representation whose dimension is $X$ and whose weight is weight $(X)$. Approximate inverse systems were first introduced by S. Mardešić and this author. The concept generalizes that of inverse system and was essentially used in proving that each Hausdorff compactum of integral cohomological dimension $\leq n$ is the cell-like image of a Hausdorff compactum of covering dimension $\leq n$.