A Compendium of Continuous Lattices
The purpose of this monograph is to present a fairly complete account of the development of the theory of continuous lattices as it currently exists. An attempt has been made to keep the body of the text expository and reasonably self-contained; somewhat more leeway has been allowed in the exercises. Much of what appears within the text constitutes basic, foundational or elementary material needed for the theory, but a considerable amount of more advanced exposition is also included.
Chapter O A Primer of Complete Lattices 1 Generalities and notation
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algebraic lattices closure operator compact Hausdorff compact pospace compact semilattices compact topological semilattice complete Heyting algebras complete lattice completely distributive lattices containing continuous lattice continuous poset continuous semilattice convergence convex COROLLARY define definition denote directed set distributive continuous lattices dual duality EXERCISE exists F-algebra finite infs following statements full subcategory functor Hence HINT Hofmann homomorphisms implies interpolation property intersection irreducible isomorphism Lawson topology LEMMA lower adjoint lower sets maps Math meet-continuous lattice monotone morphisms nonempty open filter open sets open upper sets partial order patch topology preserves arbitrary infs preserves directed sups preserving arbitrary sups prime elements prime ideal Proof PROPOSITION Remark satisfies Scott continuous Scott topology Scott-open sets Section semicontinuous semilattice with small small semilattices sober space Spec statements are equivalent subsemilattice sup-closed Suppose surjective Theorem theory topological lattice topological space ultrafilter upper adjoint upper set way-below relation