A lower bound for the dimension of certain 𝐺_{𝛿} sets in completely normal spaces

Abstract

Nagami and Roberts have proved [3, Theorem l] that if X is a normal space of dimension at least n satisfying certain conditions, then dim (X — U?TM 1 Ai)^n — i if the Ai are disjoint closed subsets of X. In this paper we allow the Ai to intersect provided that the dimension of the pairwise intersections is known. (The dimension of the remainder is reduced accordingly.) By dimension (denoted dim) we mean the covering dimension. The symbols bdy and int are used to denote the topological boundary and interior. We will say that a Ti space X is w-solid (n a positive integer) if for every point x of X and every open neighborhood U of x there is a connected open neighborhood V of x such that VQU, F is compact, dim V^n, and no (n — 2)-dimensional subset of V separates it. Such a space is locally compact and locally connected. This property is hereditary on open subsets.