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

DOI:
https://doi.org/10.1090/S0002-9939-08-09502-6

Published electronically:
October 8, 2008

MathSciNet review:
2448596

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a complete topological 2-manifold, 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 .

**1.**Alexander Blokh, Michał Misiurewicz, and Lex Oversteegen,*Planar finitely Suslinian compacta*, Proc. Amer. Math. Soc.**135**(2007), no. 11, 3755–3764. MR**2336592**, https://doi.org/10.1090/S0002-9939-07-08953-8**2.**Martin R. Bridson and André Haefliger,*Metric spaces of non-positive curvature*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486****3.**G. A. Brouwer,*Green's functions from a metric point of view*, Ph.D. dissertation, University of Alabama at Birmingham, 2005.**4.**Morton Brown,*Sets of constant distance from a planar set*, Michigan Math. J.**19**(1972), 321–323. MR**0315714****5.**K. Kuratowski,*Topology. Vol. II*, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR**0259835****6.**R. L. Moore,*Concerning triods in the plane and junction points of plane continua*, Proc. Nat. Acad. Sci. U.S.A.**14**(1928), 85-88.**7.**Sam B. Nadler Jr.,*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552**

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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:
https://doi.org/10.1090/S0002-9939-08-09502-6

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