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algebraic lattices auxiliary relation Chapter closure operator compact Hausdorff compact pospace compact semilattices compact topological semilattice complete Heyting algebras complete lattice completely distributive lattices conditions are equivalent contained continuous lattice continuous poset continuous semilattice convergence convex COROLLARY define definition denote directed set distributive continuous lattices duality Exercise exists finite infs following statements full subcategory functor Hence Hint Hofmann homomorphisms implies injective interpolation property intersection isomorphism jc€L Lawson topology Lemma locally quasicompact lower adjoint lower sets meet-continuous lattice monotone morphisms nonempty open filter open sets open upper set partial order preserves arbitrary infs preserves arbitrary sups preserves directed sups prime elements projective limits Proof Proposition Remark satisfies Scott continuous Scott topology Scott-open sets Section small semilattices sober space Spec statements are equivalent strict chain subsemilattice Suppose surjective T0-space Theorem topological lattice topological space ultrafilter upper adjoint upper set way-below relation whence