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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.
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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

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References

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