On the Fixed Point Property for Inverse Limits of Fans

In 1984, M. M. Marsh gave conditions under which the inverse limit of an inverse sequence of fans has the fixed point property. In this paper we give an extension of that result. 1. Definitions and notation By a space we mean a topological space. A continuum is a nonempty compact connected metric space. By a map we mean a continuous function. A fixed point of a map f : X → X is a point x ∈ X such that f(x) = x. A space X has the fixed point property if every map f : X → X has a fixed point. The symbols N and R denote the set of the positive integers and the set of the real numbers, respectively. The diameter of a set A in a metric space will be denoted by diam(A). Given a point x in a product ∏ {Xn : n ∈ N} we will write x = (xn)n∈N. An inverse sequence is a double sequence {Xn, fn} of spaces Xn and maps fn : Xn+1 → Xn. The inverse limit of the inverse sequence {Xn, fn}, which is denoted by lim ←− {Xn, fn} or by X∞, is the space of all points x ∈ ∏ {Xn : n ∈ N} such that xn = fn(xn+1). If Xn = X for each n ∈ N, then we write {X, fn} instead of {Xn, fn}. The projection map from X∞ into Xn will be denoted by f n . 2000 Mathematics Subject Classification. 54H25.