zbMATH — the first resource for mathematics

Topology via logic. Paperback ed. (English) Zbl 0922.54002
Cambridge Tracts in Theoretical Computer Science. 5. Cambridge: Cambridge University Press. xii, 200 p. (1996).
[For a review of the original edition (Cambridge, 1989) see Zbl 0668.54001.]
A frame is a poset in which every subset has a join, every finite subset has a meet, and in which the infinite distributive law $$x\wedge \bigvee Y= \bigvee\{x\wedge y$$; $$y\in Y\}$$ holds. Take a frame $$A$$ and a set $$X$$ together with a relation $$\models \subseteq S\times A$$ matching the frame properties (thus $$x\models \bigwedge S$$ iff $$\forall a\in S:x\models s$$, whenever $$S$$ is finite, and $$x\models\bigvee S$$ iff $$\exists a\in S:x\models s$$). Then $$(X,A,\models)$$ is called a topological system. The author develops in the first part an algebraic theory of frames with a particular emphasis on generators and relations, hence much in the spirit of abstract algebra. He investigates also properties of topological systems by showing what parts of traditional set-theoretic topology can be carried over, sometimes, how this can be done at all in the absence of an explicit notion of ‘set’ and ‘element’. This easily leads to Scott topology and to the well-known applications to the semantics of programming languages. This important algebraic aspect is further developed, in particular in Chapter 9 on spectral locales (i.e., topological systems that may be represented through frame homomorphisms). The latter part of the book is devoted to domain theory which is formulated through the machinery developed so far, and on power domains, a topic that seems also to be interesting from a purely topological point of view. There is also a small part thrown in from modal logics. Each chapter is accompanied by a number of well chosen exercises that are sometimes very sophisticated and provide frequently examples and – more important – counterexamples.
The style of writing is rather informal (to my taste more so than in the book by P. T. Johnstone [Stone spaces (1982; Zbl 0499.54001)] that the author frequently refers to), hence enabling the author to cover a lot on comparatively small space. Reading the book was (occasionally great) fun. I recommend it to readers who want to see with selected examples what topology may be good for in computer science. Computer scientists may be familiar with some of the examples, albeit in a different disguise, and they might be encouraged to look into the examples to see that topology not always needs a set-theoretic base to be helpful.

MSC:
 54-02 Research exposition (monographs, survey articles) pertaining to general topology 06B35 Continuous lattices and posets, applications 06D22 Frames, locales 54B30 Categorical methods in general topology 68Q55 Semantics in the theory of computing