Frames and Locales: Topology without Points

Item 151 in the bibliography to the book under review is an article by P. T. Johnstone titled nothing less than “The art of pointless thinking…” There’s a qualifier, namely, “a student’s guide to the category of locales.” So, notwithstanding the irresistible pull of wordplay and the provocative availability of puns, we are dealing with something quite respectable. In fact, Johnstone himself is acknowledged as a major figure in the subject, with his 1983 monograph, Stone Spaces, “still, after a quarter of a century, the standard reference book,” according to Picado and Pultr on their Preface. Johnstone’s book is something of a paean to the Stone Representation Theorem, more precisely the Stone Representation Theorem for Boolean algebras, which asserts that

the spaces which arise as the prime ideal spaces of their Boolean algebras of clopen subsets can be characterized in purely topological terms, as being compact, Hausdorff, and totally disconnected … [S]ubsequent authors have chosen to honour [Marshall Stone] by christening them “Stone spaces” …

In this connection, then, Picado and Pultr start off their treatment with the observation that “point-free topology is based, roughly speaking, on the fact that the abstract lattice of open sets can contain a lot of information about a topological space, and that an algebraic treatment can provide new insights into the nature of spaces.”

Returning briefly to Johnstone’s book, his Introduction contains a telling quote revealing some of Stone’s attendant philosophy: “A cardinal principle of modern mathematical research may be stated as a maxim: ‘One must always topologize.’” Evidently, almost three quarters of a century years later (Stone’s remark dates to 1938) the converse of this maxim motivates Picado and Pultr: they evidently propose to go from topology to algebra — but in a very different way than what is done in algebraic topology proper, with (co)homology being a functor from topological spaces to, typically, abelian groups. Nonetheless we’re still heavily steeped in categories and functors, of course: the book under review devotes a beefy second appendix to the subject of categories.

At the same time, however, the prevailing flavor of this material is a bit more along the lines of, for lack of a better word, set theory and logic: the first appendix (not quite as beefy, but still a serious affair) is devoted to the subject of posets and takes us from something as innocuous as Zorn’s Lemma to nothing less than Heyting algebras.

But we’re starting at the wrong end: what is the book itself about, as a function of its chapters? Well, in a total of fifteen chapters the reader is taken from spaces and lattices of open sets, through a thorough discussion of the notions of frames and locales, to a host of mainstays of topology: separation axioms, compactness and local compactness, uniformity, paracompactness, metrization, connectedness, and so on. The book ends (modulo its appendices) with a discussion of localic groups: by definition, a localic group is a group in the category of locales and localic maps, so we need to define this fundamental notion — after all, together with frames, that’s what this entire book is ultimately all about.

First, a frame is a complete lattice satisfying a certain distributivity condition, equipped with frame homomorphisms, and we get a category. Subsequently the desired category of locales arises as its opposite category (by reversing the arrows). The latter is regarded as “an extension of the category of sober spaces (and the frames — now referred to as locales — as generalized spaces)” A sober space, X, in turn, is characterized by a technical property to the effect that there are no “meet-irreducible” proper open subsets other than sets of the form X\{x}. A propos, sobriety is actually pretty common: all Hausdorff spaces are sober.

By the way, as the authors observe on page 2, the notion of sober spaces actually goes back to Grothendieck (who gave a variant of the above definition); this is apposite, given that it was he after all who could claim the lion’s share of credit for revolutionizing, first, algebraic geometry, then just about everything else, by going dramatically pointless in the sense of placing the emphasis on arrows as objects when training homological algebra on the problems he was dealing with. Yes, this business has a fine pedigree.

Finally, Frames and Locales: Topology without Points is quite well-written: it’s a clear and thorough treatment of what is really rather an important subject, which is to say, a greatly underappreciated one. Would that it would start to catch on more — the back-cover blurb includes the following: “Neglecting points, only little information [is] lost, while deeper insights [are] gained; … many results previously dependent on choice principles [become] constructive. The result is often a smoother, rather than a more entangled, theory …” To be sure.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.