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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 742428
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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