Fixed point sets of metric and nonmetric spaces
Authors:
John R. Martin and William Weiss
Journal:
Trans. Amer. Math. Soc. 284 (1984), 337353
MSC:
Primary 54H25; Secondary 03E35, 54A35
MathSciNet review:
742428
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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 selfmapping of . It is shown that connected subgroups of the plane and compact groups need not have CIP, and CIP need not be preserved by selfproducts of Peano continua, nonmetric manifolds or 0dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have CIP. In particular, an uncountable product of real lines, circles or twopoint 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:
http://dx.doi.org/10.1090/S00029947198407424281
PII:
S 00029947(1984)07424281
Keywords:
Fixed point set,
complete invariance property,
(Cartesian) product space,
topological group,
continuum hypothesis,
Martin's Axiom
Article copyright:
© Copyright 1984
American Mathematical Society
