Topologically convex spaces
by
Wladyslaw Kulpa
University of Silesia, Department of Mathematics, Bankowa 14, 40 007 Katowice, Poland

The notion of a convex set in geometry is replaced by some topological conditions which enables us to obtain extensions of the Schauder-Tychonoff fixed point theorem and the Helly theorem on the intersection of convex sets stating that if there are given m+2 convex subsets of n-dimensional Euclidean space Rⁿ with m >= n, and if each collection of n+1 of the subsets has a point in common then there is a common point of the m+2 subsets.

This theorem was discovered by Helly in 1913 and communicated by him to Radon who published a first proof in 1921. We shall present some general results involving k-connectedness of intersections of m-k sets. As corollaries we get

Theorem 1. Let C with |C| = m+2 be a family of \infty-connected (contractible) subsets of a \infty-connected (contractible) Hausdorff space X with dim X <= m , such that the intersection of each subfamily of C consisting at most of m+1 subsets is a nonempty \infty-connected (contractible) set. Then the intersection \cap C is a nonempty set.

Theorem 2. If a \infty-connected (contractible) Hausdorff space X has a base which members are \infty-connected (contractible) sets and the base is closed under finite intersections, then any continuous compact selfmap of X has a fixed point

The both Theorems can be obtained from Brouwer's fixed point theorem. In fact, Theorem 1 is equivalent to the Brouwer theorem. Some applications of Helly's theorem one can find in Chichilinsky expository article, Intersecting families of sets and the topology of cones in economics, Bulletin Amer. Math. Soc. 29(2) (1993), 189-207.

Date received: June 15, 2000