Sets of constant distance from a compact set in 2manifolds with a geodesic metric
Authors:
Alexander Blokh, Michał Misiurewicz and Lex Oversteegen
Journal:
Proc. Amer. Math. Soc. 137 (2009), 733743
MSC (2000):
Primary 54E35, 54F15
Published electronically:
October 8, 2008
MathSciNet review:
2448596
Fulltext PDF Free Access
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Additional Information
Abstract: Let be a complete topological 2manifold, possibly with boundary, with a geodesic metric . Let be a compact set. We show then that for all but countably many each component of the set of points distant from is either a point, a simple closed curve disjoint from or an arc such that consists of both endpoints of and that arcs and simple closed curves are dense in . In particular, if the boundary of is empty, then each component of the set is either a point or a simple closed curve and the simple closed curves are dense in .
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 A. Blokh, M. Misiurewicz and L. Oversteegen, Planar finitely Suslinian compacta, Proc. Amer. Math. Soc. 135 (2007), 37553764. MR 2336592
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 K. Kuratowski, Topology II, Academic Press, New York, 1968. MR 0259835 (41:4467)
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 R. L. Moore, Concerning triods in the plane and junction points of plane continua, Proc. Nat. Acad. Sci. U.S.A. 14 (1928), 8588.
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 S. B. Nadler, Jr., Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics 158, Marcel Dekker, Inc., New York, 1992. MR 1192552 (93m:54002)
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Additional Information
Alexander Blokh
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 352942060
Email:
ablokh@math.uab.edu
Michał Misiurewicz
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 462023216
Email:
mmisiure@math.iupui.edu
Lex Oversteegen
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 352942060
Email:
overstee@math.uab.edu
DOI:
http://dx.doi.org/10.1090/S0002993908095026
PII:
S 00029939(08)095026
Keywords:
Finitely Suslinian,
set of points of constant distance,
geodesic space
Received by editor(s):
February 8, 2007
Received by editor(s) in revised form:
January 3, 2008
Published electronically:
October 8, 2008
Additional Notes:
The first author was partially supported by NSF grant DMS 0456748
The second author was partially supported by NSF grant DMS 0456526
The third author was partially supported by NSF grant DMS 0405774
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2008
American Mathematical Society
