Over the last 45 years, Boolean theorem has been generalized and extended in several different directions and its applications have reached into almost every area of modern mathematics; but since it lies on the frontiers of algebra, geometry, general topology and functional analysis, the corpus of mathematics which has arisen in this way is seldom seen as a whole. In order to give a unified treatment of this rather diverse body of material, Dr Johnstone begins by developing the theory of locales (a lattice-theoretic approach to 'general topology without points' which has achieved some notable results in the past ten years but which has not previously been treated in book form). This development culminates in the proof of Stone's Representation Theorem.
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arbitrary axiom Boolean algebra C-ideal called clear clearly closed coherent colimits commutative compact Hausdorff completely regular condition consider construction contains continuous map Conversely Corollary corresponding covers deduce define definition denote directed distributive lattice duality easily easy elements embedding equivalent example Exercise exists extends fact filter finite follows frame functions functor give given Hausdorff space hence Heyting algebra holds homomorphism ideal identify identity implies inclusion induced isomorphic joins left adjoint Lemma limits locale Math maximal ideal meets morphism neighbourhood object obtain open sets operations paragraph particular points poset preserves prime ideal Proof Proposition prove regard relation representation restriction result ring satisfies semilattice sends sheaf Similarly space Stone structure studied sublocale subset suppose term Theorem theory topology union unique upper verify write