© 1996, Topology Atlas

Dimension theory came into being in the early part of this century with consideration of the question, "What topological property of Euclidean n-space makes it n-dimensional?" Early contributors to answering this question include Poincare, Lebesque, Brouwer, Urysohn, and Menger. Ultimately, three notions of dimension, ind, Ind, and dim, all of which can be broadly applied, prevailed. The functions ind and Ind arose from the idea that n-space is n-dimensional because closed sets which separate it are n-1 dimensional; dim came from the idea that n-space has arbitrarily small closed covers with no point common to n+2 of the sets.

The dimension theory of separable metric spaces (where ind = Ind =dim) developed quickly and the basic theory is given in the 1941 book "Dimension Theory" by Hurewicz and Wallman. Nevertheless, probably due to the interesting topology/geometry/algebra interplay involved, the dimension theory of separable metric spaces is still of considerable interest with many recent results involving infinite dimensions, product spaces, and compact spaces; there are connections with algebraic topology and dynamical systems.

It took longer to develop, but a good theory for dim (= Ind) was developed for the nonseparable metric case. Roy's example, in 1962, showed that dim and ind need not be the same, and there is no reasonable theory for ind. Dimension theory examples of nonseparable metric spaces tend to be complicated, and basic questions are unanswered; the most compelling is, "Is there a metric space X with dim X - ind X > 1?"

Recently, Engelking has revised and expanded his classic book on dimension theory into "Theory of dimensions, finite and infinite" which is now available (Helderman Verlag). This is the definitive reference book on dimension theory. In addition to the expected topics are three chapters on recent developments involving infinite dimensional spaces, an area of much recent research.