Fixed point sets of metric and nonmetric spaces

A space $X$ is said to have the complete invariance property $($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 CIP, and 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 CIP. In particular, an uncountable product of real lines, circles or two-point spaces has CIP. Examples are given which contrast the behavior of CIP in the nonmetric and metric cases, and examples of spaces are given where the existence of CIP is neither provable nor refutable with the usual axioms of set theory.