Fixed point sets of metric and nonmetric spaces

Abstract:

A space $X$ is said to have the complete invariance property $(\text {CIP})$ if every nonempty closed subset of $X$ is the fixed point set of some self-mapping of $X$. It is shown that connected subgroups of the plane and compact groups need not have $\text {CIP}$, and $\text {CIP}$ need not be preserved by self-products of Peano continua, nonmetric manifolds or $0$-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have $\text {CIP}$. In particular, an uncountable product of real lines, circles or two-point spaces has $\text {CIP}$. Examples are given which contrast the behavior of $\text {CIP}$ in the nonmetric and metric cases, and examples of spaces are given where the existence of $\text {CIP}$ is neither provable nor refutable with the usual axioms of set theory.