THE FIXED POINT PROPERTY OF THE PRODUCTS AND HYPERSPACES OF ARBOROIDS

Abstract

This paper is motivated by the Cauty’s result [1] which states that if $X$ is a dendroid, then $2^X$ and $C\left(X\right)$ have the fixed point property. The main purpose of this paper is to study the fixed point property of the hyperspaces of arboroids. It is proved, using the inverse systems method and Cauty’s result, that if $X$ is an arboroid and $f:2^X\to 2^X$ is a mapping, then $f$ has the fixed point property. Similar theorem it is proved for $f:C\left(X\right)\to C\left(X\right)$.