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Rigid subsets in Euclidean and Hilbert spaces

Published online by Cambridge University Press:  09 April 2009

Ludvik Janos
Ludvik Janos
Affiliation:
University of Newcastle, Newcastle N.S.W. 2308, Australia
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Abstract

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A subset Y of a metric space (X, p) is called rigid if all the distances p(y1, y2) between points y1, y2 ∈ Y in Y are mutually different. The main purpose of this paper is to prove the existence of dense rigid subsets of cardinality c in Euclidean spaces En and in the separable Hilbert space l2. Some applications to abstract point set geometries are given and the connection with the theory of dimension is discussed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 1 , August 1975 , pp. 66 - 72
Copyright
Copyright © Australian Mathematical Society 1975

References

Killgrove, R. B. (1971), ‘Point set geometry’, Math. Mag. 44, 115–119.CrossRefGoogle Scholar
Janos, Ludvik (1972), ‘A metric characterization of zero-dimensional spaces’, Proc. Amer. Math. Soc. 31, 268–270.CrossRefGoogle Scholar