Almost Continuous Multi-Maps and M-Retracts

Abstract

We give results about almost continuous multi-valued functions and a characterization of compact almost continuous $M$-retracts of the Hilbert cube $Q$, where almost continuity is in the sense of Stallings instead of Husain. For instance, each connectivity or almost continuous point to closed-set valued multi-function $f:I \to I$, where $I=[0\,,\,1]$, has a fixed point; i.e., a point $x\in I$ such that $x\in f(x)$. When $Y$ is a compact subset of $Q$, a sufficient condition is given for a continuous multifunction $r:Y\to Y$, with $x\in r(x)$ $\forall x\in Y$, to have an almost continuous multi-valued extension $r:Q \to Y$.