Fixed point sets of metric and nonmetric spaces

Authors:
John R. Martin and William Weiss

Journal:
Trans. Amer. Math. Soc. **284** (1984), 337-353

MSC:
Primary 54H25; Secondary 03E35, 54A35

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742428-1

MathSciNet review:
742428

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Abstract | References | Similar Articles | Additional Information

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.

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Keywords:
Fixed point set,
complete invariance property,
(Cartesian) product space,
topological group,
continuum hypothesis,
Martin’s Axiom

Article copyright:
© Copyright 1984
American Mathematical Society