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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 $ ($CIP$ )$ 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.

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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

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