Examples and Questions in the Theory of Fixed Point Sets

Abstract

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 ].

Locations

• Canadian Journal of Mathematics - View - PDF