Lebesgue covering dimension

Abstract

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 ℝ 2 n +1 are homeomorphisms, i.e. provide an embedding of X into ℝ 2 n +1 .