An addition theorem for n-fundamental dimension in metric compacta

Abstract

We generalize a notion of Freudenthal by proving that each metric compactum X is the inverse limit under an irreducible polyhedral representation of an extendable inverse sequence of compact triangulated polyhedra. The extendability criterion means that whenever X is a closed subspace of a metric compactum Y, then Y is the limit of an inverse sequence of polyhedra where all the bonding maps and triangulations are extensions of the one for X. We apply this to the theory of n-shape by using it to prove an addition theorem for n-fundamental ( n-Fd) dimension. The theorem states that if a metric compactum Z is the union of two closed subspaces X 1, X 2 with X 0 = X 1 ∩ X 2 and such that dim Z ⩽ n + 1, then n-Fd Z ⩽ max{ n-Fd X 1, n-Fd X 2, n-Fd X 0 + 1}.