Real Analysis Exchange

Almost Continuous Multi-Maps and M-Retracts

Harvey Rosen

Full-text: Open access

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 $r:Q \to Y$.

Article information

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

Dates
First available in Project Euclid: 29 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1256835199

Mathematical Reviews number (MathSciNet)
MR2010329

Zentralblatt MATH identifier
1195.54045

Subjects
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]

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

Citation

Rosen, Harvey. Almost Continuous Multi-Maps and M -Retracts. Real Anal. Exchange 34 (2008), no. 2, 471--482. https://projecteuclid.org/euclid.rae/1256835199


Export citation

References

  • J. Cornette, Connectivity functions, and images on Peano continua, Fund. Math., 58 (1966), 183–192.
  • R. Gibson, T. Natkaniec, Darboux like functions, Real Anal. Exchange, 22(2) (1996-1997), 492–533.
  • J. Girolo, The Schauder fixed point theorem for connectivity maps, Colloq. Math., 44 (1981), 59–64.
  • J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc., 52 (1942), 22–36.
  • K. R. Kellum, The equivalence of absolute almost continuous retracts and $\epsilon$-absolute retracts, Fund. Math., 96 (1977), 229–235.
  • R. L. Plunkett, A fixed point theorem for continuous multi-valued transformations, Proc. Amer. Math. Soc., 7 (1956), 160–163.
  • H. Rosen, Nonseparating almost continuous retracts of $I^n$, Proc. Amer. Math. Soc., 91 (1984), 118–122.
  • J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math., 2 (1930), 171–180.
  • J. Stallings, Fixed point theorems for connectivity maps, Fund. Math., 47 (1959), 249–263.
  • W. L. Strother, On an open question concerning fixed points, Proc. Amer. Math. Soc., 4 (1953), 988–993.
  • W. L. Strother, Fixed points, fixed sets, and M-retracts, Duke Math. J., 22 (1955), 551–556.
  • L. E. Ward, A fixed point theorem, Amer. Math. Monthly, 65 (1958), 271–272.
  • M. Wojdyslawski, Sur la contractilité des hyperespaces de continus localement connexes, Fund. Math., 30 (1938), 247–252.
  • M. Wojdyslawski, Rétractes absolus et hyperespaces des continus, Fund. Math., 32 (1939), 184–192.