Topological Spaces as Generalized Distance Spaces

This book is a study of interrelationships of metric spaces, uniform spaces and (general) topological spaces.

It introduces a “distance” definition generalizing the one used for metric spaces. This generalized distance supports natural generalizations of continuity and uniform continuity in the more familiar context of metric spaces.

Most constructs of general topology can be derived from these generalized distances type considerations, and they support the concept of “uniform continuity” in contexts even more general than traditional uniform spaces.

These generalized definitions permit the definition of DST, the category of distance spaces and (generalized) continuous functions, and this category turns out to be isomorphic to TOP, the category of topological spaces and (topologically) continuous functions. Hence all subcategories of TOP also determine associated subcategories of DST just as subcategories of DST will determine subcategories of TOP.

Among the DST subcategories are:

D0 distance spaces, which are associated with T0 topological spaces.

DR0 distance spaces, which are associated with R0 topological spaces.

D1 distance spaces, which are associated with T1 topological spaces.

Divisible distance spaces, which are associated with regular topological spaces, and also with locally uniform spaces.

Uniformly divisible distance spaces, which are associated with completely regular topological spaces, and also with uniform spaces.