Ii Locales

This chapter is an introduction to the basic concepts, constructions, and results concerning locales. Locales (frames) are the object of study of the so called pointfree topology. They sufficiently resemble the lattices of open sets of topological spaces to allow the treatment of many topological questions. One motivation for the theory of locales is building topology on the intuition of “places of non-trivial extent” rather than on points. Not the only one; hence it is not surprising that the theory has developed beyond the purely geometric scope. Still, we can think of a locale as of a kind of space, more general than the classical one, allowing us to see topological phenomena in a new perspective. Other aspects are, for instance, connections with domain theory [53, 52], continuous lattices [5, 31], logic [65, 20] and topos theory [42, 20].