Real Analysis Exchange

Almost Continuous Multi-Maps and M-Retracts

Harvey Rosen

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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 $r:Q \to Y$.

Article information

Real Anal. Exchange, Volume 34, Number 2 (2008), 471-482.

First available in Project Euclid: 29 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C05: Continuous maps 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20] 26E25: Set-valued functions [See also 28B20, 49J53, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx} 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]

$M$-retracts fixed points continuous connectivity almost continuous multi-valued functions


Rosen, Harvey. Almost Continuous Multi-Maps and M -Retracts. Real Anal. Exchange 34 (2008), no. 2, 471--482.

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