Fixed point sets of Peano continua

Abstract

1* Introduction* A subset A of a topological space X is called a fixed point set of X if there is a (continuous) map /: X—>X such that f(x) = x iff x 6 A. If X is Hausdorff, then A is closed, and, clearly, every retract of X is a fixed point set of X. It is possible that a space X may have the property that each of its nonempty closed subsets is a fixed point set of X. The problem of determining which spaces have this property, called the complete invariance property by L. E. Ward, Jr. in [5], has been investigated by H. Bobbins, Helga Schirmer, and L. E. Ward, Jr. Some spaces known to have the complete invariance property include w-cells [1], dendrites [2], convex subsets of Banach spaces [5], compact manifolds without boundary [3], and all compact triangulable manifolds with or without boundary [4]. The general question as to what properties a space must satisfy to insure that it has the complete invariance property has not been resolved. In fact, in [5, p. 553] L. E. Ward, Jr. asks the following question. Does every Peano continuum have the complete invariance property*! The purpose of this note is to show that even acyclic Peano continua which possess higher order local connectedness need not have the complete invariance property. Indeed, for each positive integer n = 1, 2, , we give an example of an (n + l)-dimensional acyclic LC~ continuum Xn which fails to have the complete invariance property. Moreover, Xn contains an ^-dimensional sphere which is not a fixed point set of Xn.