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Spaces of largest Hausdorff dimension

Part of: Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  26 February 2010

Christoph Bandt
Christoph Bandt
Affiliation:
Mathematics Department, Addis Ababa University, POB 1176, Addis Ababa, Ethiopia.
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Extract

A metric space (X, ρ) is called precompact, if, for every ε > 0, there is a finite ε-cover (a covering by sets of diameter ≤ ε). The space (X, ρ) is separable if for every e there is a countable ε-cover. There should be some in-between condition. We say that (X, ρ) has fine covers, if, for every ε > 0, there exists a countable ε-cover (U1, U2, …), such that the diameter ∂(Ui) tends to zero as i → ∞. In fact, Goodey [1] has related this property to Hausdorff dimension. We show that a space with fine covers need not be σ-precompact and that on any complete metrizable non-σ-compact space X there is a metric ρ* such that (X, ρ*) has no fine cover.

MSC classification

Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Mathematika , Volume 28 , Issue 2 , December 1981 , pp. 206 - 210
Copyright
Copyright © University College London 1981

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References

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