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 | References | Similar Articles | Additional Information

Abstract: A space is said to have the complete invariance property CIP if every nonempty closed subset of is the fixed point set of some self-mapping of . 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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Additional Information

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

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

Article copyright:
© Copyright 1984
American Mathematical Society