Topology via logic. (English) Zbl 0668.54001

Vickers, Steven

Cambridge Tracts in Theoretical Computer Science, 5. Cambridge etc.: Cambridge University Press. 200 p., £20.00; $ 39.50 (1989).

Topology (or geometric logic) is the logic of finite observations. So, the author uses locales to introduce topology. Axioms for topology constitute logic (finite conjunctions and arbitrary disjunctions). This justifies the title of the book.
After dealing with frames, topological systems are introduced. Point logic, the Scott open filter theorem, spectral algebraic locales, domain theory, power domains and spectra of rings are the topics given in this book.
A topological space is said to be sober iff every irreducible closed set has a unique generic point. A nice result given in this book is that Hausdorff spaces are sober. Also, let \(D\) be a spectral space; then \(D\) is Hausdorff iff it is a Stone space (p. 128). See the earlier book on locale theory by P. T. Johnstone [Stone spaces (1982; Zbl 0499.54001)]. Exercises, bibliography and index are available.
The get-up is attractive, and this book is a text on topology for computer scientists, because open sets are semidecidable properties.

Reviewer: K.Chandrasekhara-Rao

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

Keywords:

sober spaces; locales; frames; point logic; Scott open filter theorem; domain theory; power domains; spectra of rings; Stone space

Citations:

Zbl 0499.54001