Two Generalizations of Brouwer Fixed Point Theorem

Abstract

The following fixed point theorems are given: (1) If X is a Hausdorff and compact space and g : X → X is a oneone continuous function, then g has a fixed point. (2) If X is a compact, Hausdorff and second countable space and f : X → X is a contraction mapping, then f has a fixed point.Two proofs of Theorem 1 are given, one using sequences and the other using ultrafilters. These theorems generalize the Brouwer Fixed Point Theorem.