## Fixed point sets of metric and nonmetric spaces

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- by John R. Martin and William Weiss PDF
- Trans. Amer. Math. Soc.
**284**(1984), 337-353 Request permission

## 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.## References

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## Additional Information

- © Copyright 1984 American Mathematical Society
- 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