F-3 - Dimension Theory (General Theory)

Abstract

This chapter discusses the Dimension theory of non-metrizable spaces. This theory is a very specific part of General Topology connected with both Geometry (by means of comparisons of the topological dimension with the geometric dimension) and Algebraic Topology (by means of the cohomological dimension). Dimension theory assigns to a space either an integer, which is greater than or equal to −1 (in this case the space is called finite-dimensional), or a symbol ∞ (in this case the space is called infinite dimensional) that is named the dimension of the space. There are three basic approaches to the definition of the dimension of a space that gives three dimensions of a space ñ the covering dimension dim, the large inductive dimension Ind and the small inductive dimension ind. In the class of separable metrizable spaces (that is, in the case of the classical dimension theory), all three of these approaches are equivalent. The main goals of the Dimension theory of general spaces are the extension of the basic definitions and theorems of the classical dimension theory as far as possible, and the study of new dimensional effects arising only in general spaces.