SOME NEW CONCEPTS OF DIMENSION AND THEIR GENERALIZATION

Abstract

This chapter presents some new concepts of dimension and their generalization. In dimension theory, there are considered several concepts of dimension, two of a few more important are: Menger–Urysohn's small inductive dimension ind and Cech's great inductive dimension Ind. By their proper combination, one can get 2%0 new concepts of dimension which in turn can be still further generalized. In doing so, a fashionable pattern of modern mathematics is followed consisting in forming continuously new concepts and establishing continuously greater generalization of what is done so far, deriving a proper advantage of it. The chapter also presents the concepts of ind and Ind. The two concepts coincide for metric separable spaces but they fail to coincide even for metric spaces, for spaces between metric separable and metric. Hence, they are essentially distinct. However, both are defined by induction and one can assign to any sequence γ = (γ1, γ2, …) consisting of 0's and l's a new notion of dimension, called γ-inductive dimension and denoted γ-ind1, in a way that, in a consecutive step of calculating γ-ind X for a topological space X one follows ind if γi = 0 or Ind if γi = 1.