Fixed points in products of ordered spaces

Abstract

A topological space is said to have the fixed point property if every map (i.e. continuous function) from the space to itself leaves at least one point fixed. There is a rather well known conjecture (see e.g. [3]) asking if the fact that X and Y have the fixed point property implies that the product space X X Y has the property. This note contains an affirmative answer in the case that X and Y are compact ordered sets (with the order topology, of course). It is perhaps worth noting that there is an analogous result due to Ginsburg [1] for similarity transformations on ordered sets with the product ordered lexicographically. Since a compact ordered space has the fixed point property if and only if it is connected, we may rephrase our result to read as follows. If X and Y are compact connected ordered spaces and Z =X X Y, then Z has the fixed point property. We first establish several auxiliary properties of the space Z.