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A Theorem on Involutions on Cyclic Peano Spaces

Published online by Cambridge University Press:  20 November 2018

J. H. V. Hunt
J. H. V. Hunt*
Affiliation:
University of Virginia, Charlottesville, Virginia
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The purpose of this note is to prove that an involutionf on a cyclic Peano space S leaves some simple closed curve in Ssetwise invariant.

We shall first define the required terms. A Peano space is a locally compact, connected and locally connected metric space. A connected space is called cyclic if it has no cut-point. An involution on a space is a periodic mapping whose period is 2; it is necessarily a homeomorphism. A mapping f: X → X is said to leave a subset E of S setwise invariant if f(E) = E. These definitions may be found, for example, in [2].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 3 , September 1970 , pp. 387 - 388
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Whyburn, G. T., On the cyclic connectivity theorem, Bull. Amer. Math. Soc. 37 (1931), 429-433.Google Scholar
2. Whyburn, G. T., Analytic topology, Colloq. Publ., Vol. 28, Amer. Math. Soc., Providence, R.I., 1942.Google Scholar