Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Locally injective maps in O-minimal structures without poles are surjective

Author(s): Adam H. Lewenberg
Journal: Proc. Amer. Math. Soc. 124 (1996), 2839-2844.
MSC (1991): Primary 03C60, 06F20; Secondary 26B99, 54C30
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: If $f:\mathbf R^m\to \mathbf R^m$ is continuous and locally injective, then $f$ is in fact surjective and a homeomorphism, provided $f$ is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; `without poles' means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.

References:

[B]
N. Bourbaki, General topology, Chapters 1--4, Springer-Verlag, Berlin, 1989. MR 90a:54001a
[vdD]
Lou van den Dries, $O$-minimal structures and tame topology, preprint, 1995.
[vdDL]
Lou van den Dries and Adam H. Lewenberg, $T$-convexity and tame extensions, J. Symbolic Logic 60 (1995). MR 96a:03048
[D]
Albrecht Dold, Lectures on algebraic topology, second edition, Springer-Verlag, Berlin, 1980. MR 82c:55001
[ER]
B. Curtis Eaves and Uriel G. Rothblum, Dines-Fourier-Motzkin quantifier elimination and an application of corresponding transfer principles over ordered fields, Math. Programming 53 (1992), 307--321. MR 93h:03037
[F]
Otto Forster, Lectures on Riemann surfaces, Graduate Texts in Math., vol. 81, Springer-Verlag, New York, 1981, pp. 28--29. MR 83d:30046
[Hu]
Sze-Tsen Hu, Homotopy theory, Academic Press, New York, 1959, pp. 104--105. MR 21:5186
[KPS]
Julia F. Knight, Anand Pillay, and Charles Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), 593--605. MR 88b:03050b
[OR]
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970, pp. 133--138. MR 42:8686
[Pe]
Ya'acov Peterzil, A structure theorem for semibounded sets in the reals, J. Symbolic Logic 57 (1992), 779--794. MR 94a:03061
[Pi]
Anand Pillay, On groups and fields definable in $o$-minimal structures, J. Pure Appl. Algebra 53 (1988), 239--255. MR 89i:03069
[PSS]
A. Pillay, P. Scowcroft, and C. Steinhorn, Between groups and rings, Rocky Mountain J. Math. 19 (1989), 871--885. MR 91f:03076
[S]
Michael Spivak, A comprehensive introduction to differential geometry, Vol. 1, second edition, Publish or Perish, Houston, TX, 1979, p. 3. MR 82g:53003a

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03C60, 06F20, 26B99, 54C30

Retrieve articles in all Journals with MSC (1991): 03C60, 06F20, 26B99, 54C30

Additional Information:

Adam H. Lewenberg
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: adams@math.uiuc.edu

DOI: 10.1090/S0002-9939-96-03352-7
PII: S 0002-9939(96)03352-7
Keywords: Piecewise linear topology, PL-topology, o-minimal theory, o-minimal structure, proper map, surjective local homeomorphism
Received by editor(s): May 27, 1994
Received by editor(s) in revised form: March 14, 1995
Communicated by: Andreas R. Blass
Copyright of article: Copyright 1996, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google