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)

 
 

 

On partitions of plane sets into simple closed curves. II


Author: Paul Bankston
Journal: Proc. Amer. Math. Soc. 89 (1983), 498-502
MSC: Primary 54B15; Secondary 57N05
DOI: https://doi.org/10.1090/S0002-9939-1983-0715874-4
Addendum: Proc. Amer. Math. Soc. 91 (1984), 658.
MathSciNet review: 715874
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We answer some questions raised in [1]. In particular, we prove: (i) Let $ F$ be a compact subset of the euclidean plane $ {E^2}$ such that no component of $ F$ separates $ {E^2}$. Then $ {E^2}\backslash F$ can be partitioned into simple closed curves iff $ F$ is nonempty and connected. (ii) Let $ F \subseteq {E^2}$ be any subset which is not dense in $ {E^2}$, and let $ \mathcal{S}$ be a partition of $ {E^2}\backslash F$ into simple closed curves. Then $ \mathcal{S}$ has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.


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

  • [1] P. Bankston, On partitions of plane sets into simple closed curves, Proc. Amer. Math. Soc. (to appear). MR 702301 (85g:54007a)
  • [2] P. Bankston and R. McGovern, Topological partitions, Gen. Topology Appl. 10 (1979), 215-229. MR80i: 54007 MR 546095 (80i:54007)
  • [3] A. Beck, Continuous flows in the plane, Springer-Verlag, New York, 1974. MR 0500869 (58:18379)
  • [4] H. Cook, handwritten notes.
  • [5] S. H. Hechler, Independence results concerning the number of nowhere dense sets needed to cover the real line, Acta Math. Acad. Sci. Hungar. 24 (1973), 27-32. MR 0313056 (47:1611)
  • [6] K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [7] S. Mazurkiewicz, Remarque sur un théorème de M. Mullikin, Fund. Math. 6 (1924), 37-38.
  • [8] M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge Univ. Press, Cambridge, 1961. MR 0132534 (24:A2374)
  • [9] L. Zippin, The Moore-Kline problem, Trans. Amer. Math. Soc. 34 (1932), 705-721. MR 1501658

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B15, 57N05

Retrieve articles in all journals with MSC: 54B15, 57N05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0715874-4
Keywords: Topological partitions, euclidean plane, simple closed curves, continuous flows
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society