Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- PART I FINITE-DIMENSIONAL SETS
- 1 Lebesgue covering dimension
- 2 Hausdorff measure and Hausdorff dimension
- 3 Box-counting dimension
- 4 An embedding theorem for subsets of ℝN in terms of the upper box-counting dimension
- 5 Prevalence, probe spaces, and a crucial inequality
- 6 Embedding sets with dH(X − X) finite
- 7 Thickness exponents
- 8 Embedding sets of finite box-counting dimension
- 9 Assouad dimension
- PART II FINITE-DIMENSIONAL ATTRACTIORS
- Solutions to exercises
- References
- Index
1 - Lebesgue covering dimension
from PART I - FINITE-DIMENSIONAL SETS
Published online by Cambridge University Press: 10 January 2011
- Frontmatter
- Contents
- Preface
- Introduction
- PART I FINITE-DIMENSIONAL SETS
- 1 Lebesgue covering dimension
- 2 Hausdorff measure and Hausdorff dimension
- 3 Box-counting dimension
- 4 An embedding theorem for subsets of ℝN in terms of the upper box-counting dimension
- 5 Prevalence, probe spaces, and a crucial inequality
- 6 Embedding sets with dH(X − X) finite
- 7 Thickness exponents
- 8 Embedding sets of finite box-counting dimension
- 9 Assouad dimension
- PART II FINITE-DIMENSIONAL ATTRACTIORS
- Solutions to exercises
- References
- Index
Summary
There are a number of definitions of dimension that are invariant under homeomorphisms, i.e. that are topological invariants – in particular, the large and small inductive dimensions, and the Lebesgue covering dimension. Although different a priori, the large inductive dimension and the Lebesgue covering dimension are equal in any metric space (Katětov, 1952; Morita, 1954; Chapter 4 of Engelking, 1978), and all three definitions coincide for separable metric spaces (Proposition III.5 A and Theorem V.8 in Hurewicz & Wallman (1941)). A beautiful exposition of the theory of ‘topological dimension’ is given in the classic text by Hurewicz & Wallman (1941), which treats separable spaces throughout and makes much capital out of the equivalence of these definitions. Chapter 1 of Engelking (1978) recapitulates these results, while the rest of his book discusses dimension theory in more general spaces in some detail.
This chapter concentrates on one of these definitions, the Lebesgue covering dimension, which we will denote by dim(X), and refer to simply as the covering dimension. Among the three definitions mentioned above, it is the covering dimension that is most suitable for proving an embedding result: we will show in Theorem 1.12, the central result of this chapter, that if dim(X) ≤ n then a generic set of continuous maps from X into ℝ2n+1 are homeomorphisms, i.e. provide an embedding of X into ℝ2n+1.
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- Dimensions, Embeddings, and Attractors , pp. 7 - 19Publisher: Cambridge University PressPrint publication year: 2010