The Fundamentals of Dimension Theory

1. Brief Historical Sketch. Together with the theory of continua, dimension theory is the oldest branch of general topology. The first concepts and facts predate Hausdorff’s definition in 1914 of general Hausdorff topological spaces and, so, involved only subsets of Euclidean spaces. In its infancy, dimension theory was nurtured by the work of three outstanding mathematicians: Poincare, Brouwer, and Lebesgue. Peano’s construction in 1890 of a continuous map of a segment onto a square gave rise to the problem of whether the dimension of Euclidean space was a topological invariant. This problem was solved by Brouwer in 1911 (see the article [B1]) using the concept he introduced of the degree of a map. In the same paper Brouwer proved that, for e < 1/2, the Euclidean cube I n could not be mapped by an e-shift to a nowhere dense subset A of I n (we now know that such a set A has dimension less than n).