Helly and Klee type intersection theorems for finitary connected paved spaces

Abstract

In the present paper, the concept of n-ary and finitary connectedness is introduced, where 1-ary connectedness coincides with the usual notion of (abstract) connectedness. Relationships between ( n-ary) connectedness and an abstract concept of separation are studied. As applications, the classical intersection theorems of Helly, Klee, and others are obtained from the previous results by showing that the paving of closed convex respectively open convex subsets of a topological vector space are finitary connected. Based on a general minimax theorem, an abstract separation theorem is proved, generalizing the classical separation theorem for convex compact subsets of a locally compact topological vector space. This theorem and other results on abstract separation can be used to derive fairly general results on finitary connectedness which can be applied to various types of (convex) topological spaces.