Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A two-color theorem for analytic maps in $ {\bf R}\sp{n}$


Author: Bryan E. Cain
Journal: Proc. Amer. Math. Soc. 39 (1973), 261-266
MSC: Primary 05C15; Secondary 55A15
DOI: https://doi.org/10.1090/S0002-9939-1973-0317981-0
MathSciNet review: 0317981
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: $ {f_1}, \cdots ,{f_k}$ are real analytic functions on $ {R^n}$ th then the connected components of $ {R^n}\backslash [f_1^{ - 1}(0) \cup \cdots \cup f_k^{ - 1}(0)]$ can be ``colored'' with two colors so that two components will have different colors whenever their common boundary contains a topological $ (n - 1)$-manifold.


References [Enhancements On Off] (What's this?)

  • [1] A. S. Besicovitch, On Crum's problem, J. London Math. Soc. 22 (1947), 285-287; ibid. 23 (1948), 240. MR 9, 605; 735. MR 0025147 (9:605a)
  • [2] H. Cartan, Variétés analytiques réeles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77-99. MR 20 #1339. MR 0094830 (20:1339)
  • [3] S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952. MR 14, 398. MR 0050886 (14:398b)
  • [4] R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N.J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [5] R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math., no. 25, Springer-Verlag, Berlin and New York, 1966. MR 36 #428. MR 0217337 (36:428)
  • [6] O. Ore, The four-color problem, Pure and Appl. Math., vol. 27, Academic Press, New York, 1967. MR 36 #74. MR 0216979 (36:74)
  • [7] H. Tietze, Über das Problem der Nachbargebiete im Raum, Monatsh. Math. Phys. 16 (1905), 211-216. MR 1547413

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 05C15, 55A15

Retrieve articles in all journals with MSC: 05C15, 55A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0317981-0
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society