Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T21:40:10.530Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

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.Goodey, P. R.. “Generalized Hausdorff dimension”, Mathematika, 17 (1970), 324–327.CrossRefGoogle Scholar
2.Hausdorff, F.. “Dimension und äusseres Mass”, Math. Ann., 79 (1919), 157–179.CrossRefGoogle Scholar
3.Hurewicz, W. and Wallman, H.. Dimension theory (Princeton, 1941).Google Scholar
4.Kahnert, D.. “Hausdorff-Masse von Summenmengen I”, J. reine angew. Math., 264 (1973), 1–23.Google Scholar
5.Larman, D. G.. “A new theory of dimension”, Proc. London Math. Soc. (3), 17 (1967), 178–192.CrossRefGoogle Scholar
6.Rogers, C. A.. Hausdorff measures (Cambridge, 1970).Google Scholar
7.Wegmann, H.. “Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume”, J. reine angew. Math., 246 (1971), 46–75.Google Scholar