Topology via Logic (Cambridge Tracts in Theoretical Computer Science)

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Summary: Introducing locales, the algebraic equivalent of topologies.
Review: This is a very accessible account relating the theory of locales to classical point set topology, with remarks along the way about the relation between the two theories in terms of category theory.

Locales are lattices, which also have all joins of arbitrary subsets, just as topologies also have finite intersections of open sets and arbitrary unions; finite meets also distribute over arbitrary joins.

In locale theory, you can do topology without using points (the art of "pointless thinking", as Johnstone likes to call it), although you can also introduce points in locales by definition.

Locales, in other words, are complete lattices, and their elements can be identified with open sets. In the other direction, "points" in locales (which can be identified with principal prime ideals) correspond to the points in a corresponding topological space.

The book is a nice introduction to Johnstone's now classic *Stone Spaces*. But the "logic&qu! ot; as in *Topology via Logic*-- refers to "logic" in the sense of computer science. Readers at home with symbolic logic may therefore find the examples, all based on "data streams" of bits, a bit strange, and may want to translate them into more familiar terms. But the reward is worth the effort.

The informal style and free use of pictures to illustrate ideas is also a refreshing, and welcome change from the rigid and unnecessarily formal style too often encountered in books on this and similar topics.