The Approximation of Compacta by Finite T₀-Spaces

Abstract

It has long been known that compact Hausdorff spaces can be approximated using finite T ₀-spaces, and that many can be represented as inverse limits of polyhedra. Here we study the relationship between these two types of representation. In Section 4, we define the concept of a calming map and show that the Hausdorff reflection of the limit of an inverse sequence of finite T ₀-spaces and calming maps is the inverse limit of their corresponding polytopes and piecewise linear maps. Thus each k -dimensional metric compactum (respectively, continuum) can be characterized as the Hausdorff reflection of the limit of an inverse sequence with calming bonding maps of finite (respectively, connected) T ₀-spaces whose dimension is k; an infinite-dimensional version of this is also found.