The main purpose of this book is to bring together a number of results concerning the embedding of ‘finite-dimensional’ compact sets into Euclidean spaces, where an ‘embedding’ of a metric space (X, ϱ) into ℝn is to be understood as a homeomorphism from X onto its image. A secondary aim is to present, alongside such ‘abstract’ embedding theorems, more concrete embedding results for the finite-dimensional attractors that have been shown to exist in many infinite-dimensional dynamical systems.

In addition to its summary of embedding results, the book also gives a unified survey of four major definitions of dimension (Lebesgue covering dimension, Hausdorff dimension, upper box-counting dimension, and Assouad dimension). In particular, it provides a more sustained exposition of the properties of the boxcounting dimension than can be found elsewhere; indeed, the abstract results for sets with finite box-counting dimension are those that are taken further in the second part of the book, which treats finite-dimensional attractors.

While the various measures of dimension discussed here find a natural application in the theory of fractals, this is not a book about fractals. An example to which we will return continually is an orthogonal sequence in an infinite-dimensional Hilbert space, which is very far from being a ‘fractal’. In particular, this class of examples can be used to show the sharpness of three of the embedding theorems that are proved here.