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Dimension Theory Via Bisector Chains

Published online by Cambridge University Press:  20 November 2018

Ludvik Janos
Ludvik Janos*
Affiliation:
Mississippi State University, Mississippi State MS 39762
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Abstract

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For two subsets Z and Y of a metric space (X, d) the set Z is said to be a bisector in Y iff Z ⊂ Y and there exist two distinct points y 1, y 2 ∊ Y such that Z = {z: d(z, y 1) = d(z, y 2) and z ∊ Y}. Considering chains of consecutive bisectors X ⊃ X 1 ⊃ … ⊃ Xk we denote by b(X, d) the maximum of their length. The topological invariant b(X) is defined as the minimum of b(X, d) taken over the set of all metrizations of X. It is proved that if X is compact then dim(X) ≤ b(X) ≤ 2 dim(X) + 1, b(X) = 0 iff dim(X) = 0 and b(X) = n implies dim(X) = n for n = 1 and ∞. The sharp result b(En ) = n for n = 1, 2, … is obtained for Euclidean space En .

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 3 , 01 September 1977 , pp. 313 - 317
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. de Groot, J., On a metric that characterizes dimension Canad. J. Math. 9 (1957), 511-514. MR 19, 874.Google Scholar
2. Janos, L., A metric characterization of zero-dimensional spaces, Proc. Amer. Math. Soc. 31, (1972), 268-270.Google Scholar
3. Janos, L., Rigid subsets in Euclidean and Hilbert spaces, J. Austr. Math. Soc. 20A (1975), 66-72.Google Scholar
4. Nagata, J., On a relation between dimension and metrization, Proc. Japan Acad. 32 (1956), 237-240. MR 19, 156.Google Scholar
5. Nagata, J., Modern dimension theory, Interscience Publishers, New York, 1965.Google Scholar