Page 13 - Textos de Matemática Vol. 41

Preface xi
Of course, just having (special) mappings instead of formal morphisms would not count for all that much. But it turns out that such representation is often fairly apt, and thus opened perspective often surprisingly simplifies the techniques. In the present text we have chosen several topics of elementary point-free topology that illustrate the benefits of this approach. Needless to say, such “covariant technique” is not preferable to the “algebraic” one under all cir- cumstances (and even in this text we are not unduly consequent, and sometimes slip into the contravariant thinking if it is simpler). Therefore, our topics cover only a small part of the elementary point-free topology. But there are cases in which we gain a lot. For instance, thus defined localic maps f : L → M send spectral points (that is, completely prime elements) of L to spectral points of M and make the spectrum continuous maps PtL → PtM simply restrictions of f. We are convinced that the spectrum adjunction, and the relation between classical spaces and the more general point-free ones becomes more lucid.
In particular, however, we see the striking technical advantages (and fac- tual ones as well) when working with sublocales (generalized subspaces). They now become really sub-spaces, that is, precisely the subsets the embeddings of which are localic maps (and, moreover, they have a very simple intrinsic char- acterization). The structure of the system of sublocale is now fairly transparent (the meets are the intersections and the joins are described by a simple and ex- pedient formula), which results, a.o., in a very short proof of the distributivity. The formula for closure is extremely plain (resulting in the characterization of dense sublocales as those containing 0, and in a three-line proof of the Isbell’s density theorem). Working with images and preimages is easy (images are the set-theoretical images, and to obtain preimages the set-theoretical ones have to be only slightly modified). The analysis of separation properties (in particular of fitness, subfitness and the strong Hausdorff property) is not only technically simpler than the usual ones, but also brings new insight to some of their aspects. Also we gain some new insight into products of locales (coproducts of frames) — here, not surprisingly, our techniques are mixed.
The text does not go much beyond the topics just mentioned (and misses very important questions, even fundamental ones, like those of compactness and local compactness, of cover based uniformity and nearness, etc.). Nevertheless, it can serve as a first introduction to point-free thinking; afterwards, the reader is encouraged to open some more comprehensive text like for instance [35], or more recent but less extensive [53] and [57].
Coimbra, April 2008