This booklet (comprising 129 pages) is a readable and concise introduction to the classical theory of topological (covering) dimension, completed with a treatment of a new concept of the mean topological dimension of a dynamical system. The first chapter introduces basic concepts and it shows that in normal spaces the dimension of a countable union of closed subspaces is the supremum of their dimensions. The second chapter shows that in Hausdorff spaces (called ‘les espaces séparés’) there is a sequence of concepts of increasing strength: (covering) dimension zero; existence of a clopen base (les espaces éparpillés); totally separated; totally disconnected (these concepts are all equivalent in compact Hausdorff spaces). Chapter 3 treats the dimension of polyhedra and proves the Lebesgue lemma, which implies dim Rn = n. Chapter 4 shows that any compact metric space of dimension n can be embedded into R2n+1. Chapter 5 presents counter-examples of Knaster-Kuratowski and Tichonov.
The rest of the book treats the theory of mean topological dimension introduced recently by Gromov. The concept works for dynamical systems (self-homeomorphisms) of a normal topological space. It is defined analogously to topological entropy but the size of degree is used instead of the power of the cover. If the topological dimension of the underlying space is finite, the mean topological dimension is zero, so the theory is intended for infinite-dimensional dynamical systems. Chapter 6 introduces basic notions and chapter 7 shows that the mean topological dimension can be any nonnegative real number or infinity. Chapter 8 proves the Javorski theorem, which says that each dynamical system without periodic points on a compact metric space of a finite dimension is conjugated to a subshift of RZ. The counter-example of Lindenstrauss-Weiss shows that the assumption of the finite topological dimension is necessary.