The point of pointless topology

Abstract

Introduction. A celebrated reviewer once described a certain paper (in a phrase which never actually saw publication in Mathematical Reviews) as being concerned with the study of measures on spaces. This article contains nothing about measures, valueless or otherwise; but I hope that by giving a historical survey of the subject known as (i.e. the study of topology where open-set lattices are taken as the primitive notion) I shall succeed in convincing the reader that it does after all have some point to it. However, it is curious that the point (as I see it) is one which has emerged only relatively recently, after a substantial period during which the theory of spaces has been developed without any very definite goal in view. I am sure there is a moral here; but I am not sure whether it shows that pointless abstraction for its own sake is a good thing (because it might one day turn out to be useful) or a bad thing (because it tends to obscure whatever point there might be in a subject). That much I shall leave for the reader to decide. This article is in the nature of a trailer for my book Stone spaces [35], and detailed proofs of (almost) all the results stated here will be found in the book (together with a much fuller bibliography than can be accommodated in this article). However, I should make it plain that I do not claim personal credit for more than a small proportion of these results, and that my own understanding of the nature of topology has been enriched by my contacts with a number of other mathematicians, amongst whom I should particularly mention Bernhard Banaschewski, Michael Fourman, Martin Hyland, John Isbell, Andre Joyal and Myles Tierney. I should also mention the work of Bill Lawvere, particularly as reported in [41], on the nature of continuous variation and the conceptual relation between constant and variable quantities, which has had a profound influence on the developments which I wish to describe; but such questions as these will not be explicitly considered in the present article.