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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sets of constant distance from a compact set in 2-manifolds with a geodesic metric


Authors: Alexander Blokh, Michał Misiurewicz and Lex Oversteegen
Journal: Proc. Amer. Math. Soc. 137 (2009), 733-743
MSC (2000): Primary 54E35, 54F15
Published electronically: October 8, 2008
MathSciNet review: 2448596
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Abstract: Let $ (M, d)$ be a complete topological 2-manifold, possibly with boundary, with a geodesic metric $ d$. Let $ X\subset M$ be a compact set. We show then that for all but countably many $ \varepsilon$ each component of the set $ S(X, \varepsilon)$ of points $ \varepsilon$-distant from $ X$ is either a point, a simple closed curve disjoint from $ \partial M$ or an arc $ A$ such that $ A\cap\partial M$ consists of both endpoints of $ A$ and that arcs and simple closed curves are dense in $ S(X, \varepsilon)$. In particular, if the boundary $ \partial M$ of $ M$ is empty, then each component of the set $ S(X, \varepsilon)$ is either a point or a simple closed curve and the simple closed curves are dense in $ S(X, \varepsilon)$.


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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email: ablokh@math.uab.edu

Michał Misiurewicz
Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: mmisiure@math.iupui.edu

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email: overstee@math.uab.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09502-6
PII: S 0002-9939(08)09502-6
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