The aim of this book is to promote the fibrewise viewpoint, particularly in topology. I set out to show that by adopting this viewpoint systematically two desirable objectives can be achieved. First one can reorganize certain portions of the literature and see some well-known material in a new and clearer light. Secondly one is led on to further developments which might not have suggested themselves otherwise.

The fibrewise viewpoint is standard in the theory of fibre bundles, and some of the ideas in this volume originated in work on the foundations of that theory. However, it has been recognized only recently that the same viewpoint is also of great value in other theories, such as general topology. In fact there appear to be fibrewise versions of many of the major concepts of general topology, some of which can already be found in the literature. Having established this I go on to show that there are also fibrewise versions of uniform theory and of homotopy theory. The latter may be well known but has not, as far as I am aware, been written up before. There are important links between fibrewise topology and equivariant topology, a subject which has seen great research activity over the past ten or twenty years. I predict a growing interest in research on fibrewise topology and hope that the present volume will help to provide a foundation for such work.

In fibrewise topology we work over a topological base space B, say. When B is a point-space the theory reduces to that of ordinary topology. A fibrewise topological space over B is just a topological space X together with a continuous projection p:X → B.