No continuum in $E^ 2$ has the TMP. I. Arcs and spheres
HTML articles powered by AMS MathViewer
- by L. D. Loveland
- Proc. Amer. Math. Soc. 110 (1990), 1119-1128
- DOI: https://doi.org/10.1090/S0002-9939-1990-1027099-X
- PDF | Request permission
Abstract:
A subset $X$ of the Euclidean plane ${E^2}$ is said to have the triple midset property (TMP) if, for each pair of points $x$ and $y$ of $X$, the perpendicular bisector of the segment joining $x$ and $y$ intersects $X$ at exactly three points. In this paper it is proved that no arc or simple closed curve in ${E^2}$ can have the TMP. In a subsequent paper these results are used to prove that no planar continuum can have the TMP.References
- F. Bagemihl and P. Erdös, Intersections of prescribed power, type, or measure, Fund. Math. 41 (1954), 57–67. MR 62807, DOI 10.4064/fm-41-1-57-67
- Anthony D. Berard Jr., Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1–7. MR 295300, DOI 10.4064/fm-73-1-1-7
- A. D. Berard Jr. and W. Nitka, A new definition of the circle by the use of bisectors, Fund. Math. 85 (1974), no. 1, 49–55. MR 355991, DOI 10.4064/fm-85-1-49-55
- D. G. Larman, A problem of incidence, J. London Math. Soc. 43 (1968), 407–409. MR 231724, DOI 10.1112/jlms/s1-43.1.407
- L. D. Loveland, The double midset conjecture for continua in the plane, Topology Appl. 40 (1991), no. 2, 117–129. MR 1123257, DOI 10.1016/0166-8641(91)90045-N —, No continuum in ${E^2}$ has the TMP; II. Triodic continua, Rocky Mountain J. Math. (to appear).
- L. D. Loveland and S. G. Wayment, Characterizing a curve with the double midset property, Amer. Math. Monthly 81 (1974), 1003–1006. MR 418059, DOI 10.2307/2319308 S. Mazurkiewicz, Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, C. R. Acad. Sci. et Letters de Varsovie 7 (1914), 382-383.
- Sam B. Nadler Jr., An embedding theorem for certain spaces with an equidistant property, Proc. Amer. Math. Soc. 59 (1976), no. 1, 179–183. MR 410686, DOI 10.1090/S0002-9939-1976-0410686-X
- J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446–452. MR 355791, DOI 10.1090/S0002-9939-1975-0355791-0
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1119-1128
- MSC: Primary 54F15; Secondary 54F50
- DOI: https://doi.org/10.1090/S0002-9939-1990-1027099-X
- MathSciNet review: 1027099