Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T14:41:17.696Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension Theory via Reduced Bisector Chains

Published online by Cambridge University Press:  20 November 2018

Ludvik Janos
Ludvik Janos*
Affiliation:
Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (X, d) be a metric space and Y and Z subsets of X. We say that Z is a bisector in Y and write Y⊳Z iff Y⊃Z and there are two distinct points y1, y2 ∈ Y such that Z = ={z:d(z, y1) = d(z, y2) and z∈Y}. By a reduced bisector chain in (X, d) of length n we understand a chain X = such that dim Xn≤0 and dimXn-1>0). By r(X, d) we denote the maximum length of reduced bisector chains in (X, d). For a metrizable topological space X we introduce the topological invariant r(X) as the minimum of r(X, d) taken over the set of all metrizations d of X. We prove that the function r(X) coincides with the dimension of X on the class of compact metric spaces.

Keywords

54 F 4555 C 1054 E 35Bisectorbisector-chaindimensionmetrizationtopological invariantexpressability of a topological property in a suitable language
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 305 - 311
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. J. de Groot, On a metric that characterizes dimension, Can. J. Math. 9 (1957) 511-514.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., Dimension theory via bisector chains, Canad. Math. Bull. 20 (1977), 313-317.Google Scholar
4. Nagata, J., Modern dimension theory, Interscience Publishers, New York, 1965.Google Scholar
5. Roberts, J. H., A theorem on dimension, Duke Math. J. 8 (1941), 565-574.Google Scholar