Rocky Mountain Journal of Mathematics

No Continuum in $\bf E^2$ Has the TMP; II. Triodic Continua

L.D. Loveland

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Rocky Mountain J. Math., Volume 22, Number 3 (1992), 957-971.

First available in Project Euclid: 5 June 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54F50: Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] 54F15: Continua and generalizations
Secondary: 54F65: Topological characterizations of particular spaces 51K05: General theory 51M05: Euclidean geometries (general) and generalizations

Arc bisector continuum Euclidean plane equidistant set midset simple closed curve triod triple midset property


Loveland, L.D. No Continuum in $\bf E^2$ Has the TMP; II. Triodic Continua. Rocky Mountain J. Math. 22 (1992), no. 3, 957--971. doi:10.1216/rmjm/1181072708.

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  • F. Bagemihl and P. Erdős, Intersections of prescribed power, type, or measure, Fund. Math. 41 (1954), 57-67.
  • A.D. Berard, Jr. and W. Nitka, A new definition of the circle by use of bisectors, Fund. Math. 85 (1974), 49-55.
  • L.D. Loveland, The double midset conjecture for continua in the plane, Top. Applications 40 (1991), 117-129.
  • --------, No continuum in $E^2$ has the TMP; I. Arcs and spheres, Proc. Amer. Math. Soc. 110 (1990), 1119-1128.
  • L.D. Loveland and S.G. Wayment, Characterizing a curve with the double midset property, Amer. Math. Monthly 81 (1974), 1003-1006.
  • S. Mazurkiewicz, Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, C.R. Sc. et Letters de Varsovie 7 (1914), 382-383.
  • R.L. Moore, Foundations of point set theory, revised ed., Amer. Math. Soc. Colloq. Publ. 13, 1962.
  • Sam B. Nadler, Jr., An embedding theorem for certain spaces with an equidistant property, Proc. Amer. Math. Soc. 59 (1976), 179-183.
  • J.B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446-452.
  • Stephen Willard, General topology, Addison-Wesley, Reading, MA, 1970.