On multivalued fixed-point free maps on $\mathbb {R}^n$
HTML articles powered by AMS MathViewer
- by Raushan Z. Buzyakova PDF
- Proc. Amer. Math. Soc. 140 (2012), 2929-2936 Request permission
Abstract:
To formulate our results let $f$ be a continuous multivalued map from $\mathbb R^n$ to $2^{\mathbb R^n}$ and $k$ a natural number such that $|f(x)|\leq k$ for all $x$. We prove that $f$ is fixed-point free if and only if its continuous extension $\tilde f:\beta \mathbb R^n\to 2^{\beta \mathbb R^n}$ is fixed-point free. If one wishes to stay within metric terms, the result can be formulated as follows: $f$ is fixed-point free if and only if there exists a continuous fixed-point free extension $\bar f: b\mathbb R^n\to 2^{b\mathbb R^n}$ for some metric compactificaton $b\mathbb R^n$ of $\mathbb R^n$. Using the classical notion of colorablity, we prove that such an $f$ is always colorable. Moreover, a number of colors sufficient to paint the graph can be expressed as a function of $n$ and $k$ only. The mentioned results also hold if the domain is replaced by any closed subspace of $\mathbb R^n$ without any changes in the range.References
- Raushan Z. Buzyakova, Fixed-point free maps on the reals and more, Topology Appl. 156 (2008), no. 2, 465–472. MR 2475129, DOI 10.1016/j.topol.2008.07.020
- R. Z. Buzyakova and A. Chigogidze, Fixed-point free maps of Euclidean spaces, Fund. Math. 212 (2011), no. 1, 1–16. MR 2771585, DOI 10.4064/fm212-1-1
- N. G. de Bruijn and P. Erdös, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 369–373. MR 0046630
- Eric K. van Douwen, $\beta X$ and fixed-point free maps, Topology Appl. 51 (1993), no. 2, 191–195. MR 1229715, DOI 10.1016/0166-8641(93)90152-4
- Ryszard Engelking, General topology, Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60], PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author. MR 0500780
- M. Katětov, A theorem on mappings, Comment. Math. Univ. Carolinae 8 (1967), 431–433. MR 229228
- E. V. Ščepin, Topology of limit spaces with uncountable inverse spectra, Uspehi Mat. Nauk 31 (1976), no. 5 (191), 191–226 (Russian). MR 0464137
- Jan van Mill, The infinite-dimensional topology of function spaces, North-Holland Mathematical Library, vol. 64, North-Holland Publishing Co., Amsterdam, 2001. MR 1851014
Additional Information
- Raushan Z. Buzyakova
- Affiliation: Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, North Carolina 27402
- Email: rzbouzia@uncg.edu
- Received by editor(s): November 22, 2010
- Received by editor(s) in revised form: March 7, 2011
- Published electronically: December 13, 2011
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2929-2936
- MSC (2010): Primary 54H25, 58C30, 54B20
- DOI: https://doi.org/10.1090/S0002-9939-2011-11166-3
- MathSciNet review: 2910778