Properties of fixed point spaces

Abstract

In this note relations between the fixed point property and compactness are studied and it is shown that the fixed point property need not be preserved under the cross product. Most known theorems concerning the fixed point property (written as the f.p.p.) are for compact spaces. It is to be shown that, in the general case, compactness and the f.p.p. are only vaguely related. Example 1 is an example of a Hausdorff space that has no compact subsets except finite sets. By the use of Theorem 1, it will be shown that this space has the f.p.p. Theorem 2 states a weak compactness condition that metric spaces with the f.p.p. must possess-namely that every infinite chain of arcs must have a nonvoid limiting set. The space in Example 2 has the f.p.p., yet its cross product with itself does not satisfy this arccompactness condition, and thus cannot have the f.p.p. Example 3 is an example of a locally contractible metric space which has the f.p.p. yet is not compact. Theorem 3 states that if X is a locally connected, locally compact metric space with the f.p.p., then X is compact. Example 4 is an example of a compact metric space X which does not have the f.p.p., yet contains a dense subset Y which does have the f.p.p. Thus the f.p.p. is not preserved under closure.