Introduction

In ordinary topology one is concerned with the category Top of spaces and maps, i.e. continuous functions. In this article, however, I wish to consider rather the category TopB of spaces and maps over a given space B. My aim is to show that many of the familiar definitions and theorems of ordinary topology can be generalized, in a natural way, so that one can develop a theory of topology over a base. In fact, once the definitions have been suitably formulated the proofs of the theorems are mostly just fairly routine generalizations (see [4], Chapter 3) of those used in ordinary topology. There are, however, certain results which have no counterpart in the ordinary theory and for these, of course, I will give proofs.

A space over B, I recall, is a space X together with a map p: X → B, called the projection. Usually X alone is sufficient notation. If X is a space over B then any sub-space of X may be regarded as a space over B by restriction of the projection. Also B itself is regarded as a space over B with the identity map as projection.

If X, Y are spaces over B with projections p,q, respectively, then a map X → Y over B is a map θ: X → Y of spaces such that qθ = p. The category TopB of spaces and maps over B has various features which are relevant.