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Examples and Questions in the Theory of Fixed Point Sets

Published online by Cambridge University Press:  20 November 2018

John R. Martin  and
Sam B. Nadler Jr.
John R. Martin
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
Sam B. Nadler Jr.
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1017 - 1032
Copyright
Copyright © Canadian Mathematical Society 1979

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