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A Fixed-Point Theorem for Commuting Monotone Functions

Published online by Cambridge University Press:  20 November 2018

William J. Gray
William J. Gray*
Affiliation:
University of Alabama, University, Alabama
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Hamilton (1) proved that a hereditarily unicoherent, hereditarily decomposable metric continuum has the fixed-point property for homeomorphisms. In this paper we shall generalize this result by showing that if X is a hereditarily unicoherent, hereditarily decomposable Hausdorff continuum and 5 is an abelian semigroup of continuous monotone functions from X into X, then S leaves a point of X fixed.

Let X be a Hausdorff continuum. X is unicoherent if, whenever X = A ∪ B, where A and B are subcontinua of X, A ∩ B is a continuum. If each subcontinuum of X is unicoherent, X is hereditarily unicoherent. X is decomposable if X is the union of two of its proper subcontinua. If each subcontinuum of X which contains more than one point is decomposable, X is hereditarily decomposable.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 502 - 504
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Hamilton, O. H., Fixed points under transformations of continua which are not connected im kleinen, Trans. Amer. Math. Soc. U (1938), 18–24.Google Scholar
2. Kelley, J. L., General topology (Van Nostrand, Princeton, N.J., 1955).Google Scholar