Extendable irreducible representations of compacta

Leonard R. Rubin, Norman, Oklahoma, USA

Keywords: Inverse system, approximate inverse system, simplicially irreducible representation, dimension, nonmetrizable compactum AMS Subject classification: 54B35, 54F45

Abstract: For each infinite set ∨, we consider the class of compacta in the Tychonoff cube I^{Ω} which contain 0_{Ω}. We show that each such X can be represented extendably as the limit of an approximate (inverse) system X = (P_{a}, U_{⊣}, , √_{⊣⊣′}, (A, ≤)) of compact polyhedra where all bonding maps are piecewise

linear and irreducible and all projections are irreducible. The extendability refers to the notion that if X ⊂ Y = \overline{Y} ⊂ I^{Ω}, then there is a similar (extendable) approximate system Y = (P̃_{a}, Ũ_{a}, p̃_{aa′}, (A, ≤)) for Y where each object with the same index is an extension of the one in X.