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A connected metric space without an equally spaced chain of points

Published online by Cambridge University Press:  17 April 2009

J. Arias de Reyna
J. Arias de Reyna
Affiliation:
Facultad de Matematicas, Universidad de Sevilla, c/ Tarfia sn., Sevilla-12, Spain.
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Abstract

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We construct a connected subspace M of the euclidean plane R2 containing two points A and B such that, for every pair of points {P, Q} of M\{A, B}, the three real numbers d(A, P), d(P, Q) and d(Q, B) are not the same. This solves a question posed by Väisälä.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 2 , October 1983 , pp. 283 - 285
Copyright
Copyright © Australian Mathematical Society 1983

References

[1] Levy, A., Basic set theory (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[2] Sierpinski, W., “Sur un ensemble ponctiforme connexe”, Fund. Math. 1 (1920), 7–10.CrossRefGoogle Scholar
[3] Väisälä, J., “Dividing an arc to subarcs with equal chords”, Colloq. Math. 46 (1982), 203–204.CrossRefGoogle Scholar