The intersection properties of generalized Helly families for inverse limit spaces

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.