The intersection properties of generalized Helly families for inverse limit spaces

Abstract

The aim of this paper is to discuss the intersection properties of generalized Helly families for topological spaces and inverse limit spaces. This concept is a generalization of Helly family. A generalized Helly family C is a countable family of ∞-connected subsets of a topological space X satisfying the following conditions: the intersection ⋂E of each finite subfamily E⊂C is ∞-connected; and the intersection ⋂D of each proper subfamily D⊂C is nonempty.In [6], Kulpa (1997) extended the Helly convex-set theorem onto topological spaces in terms of Helly families. Here, we improve his result. We show that if C is a generalized Helly family of compact subsets of a topological space X and U is a countable covering of X with Cj⊂Uj, for each j∈N, then ⋂D is nonempty.