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Full Description
The notion of closure pervades mathematics, especially in the fields of topology and projective geometry. Demonstrating this pervasiveness in the field, this graduate-level book provides a complete introduction to closure systems. With an emphasis on finite spaces and algebraic closures, the text covers graph theory, ordered sets, lattices, projective geometry, and formal logic as they apply to the study of closures. Each chapter presents a vignette to illustrate the topic covered. The author also includes numerous exercises as well as concrete examples to support the material discussed.
Contents
PrefaceAn Introduction to ClosuresSix Motivating Examples Some Properties of Set MapsA Budget of ClosuresArity Interval Functions1 3-letter Scrabble Some More Algebraic ExamplesNapoleonic Closure Universality of Napoleonic ClosureBasic Properties of ClosuresClosures Systems and Their Hulls Hull Operators and the Substitution PropertyFilling in Dependencies Algebraic and Topological Closures Closures That Are Both Topological and AlgebraicIsomorphism and Cryptomorphy SeparationConstructions on Closures Small ClosuresAlignments Universal Algebras Realizing Alignments by a Single OperationUniversality of GroupoidsUniversality of Napoleonic Closure Copoints in Finitary Closures Copoints and the Separation Axioms 53 and 54Matroids and AntimatroidsIndependence More Examples of Matroids and AntimatroidsOrder Convexity Acyclic SemigroupsArticulated SumsBackground ReviewFunctionsRelationsOrders Basic Graph TheorySome More Algebraic ExamplesTwo Binary Operations Set TheoryCardinals The Axiom of ChoiceOrdinals Countability
