Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T19:33:23.301Z Has data issue: false hasContentIssue false
  • English
  • Français

Axiomatic characterisations of dimension

Part of: Special properties

Published online by Cambridge University Press:  26 February 2010

M. G. Charalambous
M. G. Charalambous
Affiliation:
Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria.
Get access

Extract

The problem of finding an axiomatic characterisation of dimension was first tackled by Menger, who gave a set of five independent axioms characterising the dimension (in the sense of dim, ind, or Ind since they are all equal on separable metric spaces) of subsets of the plane [7, p. 156]. The question of whether Menger's axioms characterise the dimension of more general spaces is still unsettled. Recently, Nishiura “11” obtained a set of seven independent axioms characterising the dimension of separable metric spaces. By modifying one of Nishiura's axioms, Aarts [1] then obtained an axiomatic characterisation of the strong inductive dimension (Ind) of metric spaces. Also, Ščepin [12] and Lokucievskiĭ [9] have obtained different axioms for dim on the class of compact metric and compact spaces, respectively. We present here four sets of independent axioms that characterise the dimension function u-dim, which is defined on the class of all uniform spaces.

MSC classification

Secondary: 54F45: Dimension theory
Type
Research Article
Information
Mathematika , Volume 23 , Issue 1 , June 1976 , pp. 84 - 88
Copyright
Copyright © University College London 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aarts, J. M.. “A characterization of strong inductive dimension”, Fund. Math., 70 (1971), 147–155.Google Scholar
2.Charalambous, M. G.. “A new covering dimension function for uniform spaces”, J. London, Math. Soc. (2), 11 (1975), 137–143.Google Scholar
3.Charalambous, M. G.. “Inductive dimension theory for uniform spaces”, Ann. Univ. Sci. Budapest. Eötvös Sect' Math., 17 (1974), 21–28.Google Scholar
4.Charalambous, M. G.. “The dimension of inverse limits”, Proc. Amer. Math. Soc. (to appear).Google Scholar
5.Charalambous, M. G.. “Uniformity-dependent dimension functions”, to appear.Google Scholar
6.Engelking, R.. Outline of General Topology (Amsterdam, 1968).Google Scholar
7.Hurewicz, W. and Wallman, H.. Dimension Theory (Princeton, New Jersey, 1948).Google Scholar
8.Isbell, J. R.. Uniform Spaces, Atner. Math. Soc, Math. Surveys No. 12 (1964).Google Scholar
9.Lokucievskil, O. V.. “Axiomatic definition of the dimension of compacta”, Soviet Math. Dokl., 14 (1973), 1477–80.Google Scholar
10.Nagami, K.. Dimension Theory (New York and London, 1970).Google Scholar
11.Nishiura, T.. “Inductive invariants and dimension theory”, Fund. Math., 59 (1966), 243–262.Google Scholar
12. E. Šĭepin. “Axiomatics of the dimension of metric spaces”, Soviet Math. Dokl., 13 (1972), 1177–1179.Google Scholar