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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adjunction algebraic category arbitrary axiom of choice bijection C-ideal clearly clopen closed coherent commutative compact Hausdorff space compact regular locale compactification complete lattice completely distributive completely regular condition consider construction contains continuous lattice continuous map continuous poset Corollary deduce define denote directed joins disjoint distributive lattice distributive law duality elements embedding equivalent Exercise exists filtered colimits follows forgetful functor frame homomorphism functions Gelfand global sections Hausdorff space hence Heyting algebra idempotent implies inclusion Ind-C intersection interval topology Isbell isomorphic KHausSp left adjoint Lemma Let locally compact Mac Lane Math maximal ideal meet-semilattice monad morphism object open neighbourhood open sets order-preserving partial order points presheaf prime filter prime ideal principal ideal Proof Let Proposition Let realcompact satisfies Scott topology Scott-open semilattice sheaf sober space spec Stone space sublocale subset subspace suppose surjective theory ultrafilter unique Zariski