Topology Proceedings

Abstract

ABSTRACT. We show that for any continua X and Y the smoothness of either the hyperspace C(X) or 2 x or of the Cartesian product X x Y implies the ptoperty of Kelley for X. An example is constructed showing that the converse is not true. A continuum is a compact connected metric space. Given a point p E X and a positive number r we denote by Bx(p, r) the open ball with center p and radius r and, for A c X we defi~e Nx(A,r) == U{B(x,r): x E A}. We say that continuum X has the property of J<elley if for each £ > 0 there is a 8 > 0 such that for each point x EX, for 1991 Mathematics Subject Classification. 54B10, 54B20, 54F15. !(ey 1vords and phrases. continuum, hyperspace, product, property of Kelley, sll10oth