Introduction to Closure Systems

Robert E. Jamison, Clemson University, South Carolina, USA

Preface


An Introduction to Closures
Six Motivating Examples
Some Properties of Set Maps
A Budget of Closures
Arity
Interval Functions
1 3-letter Scrabble
Some More Algebraic Examples
Napoleonic Closure
Universality of Napoleonic Closure


Basic Properties of Closures
Closures Systems and Their Hulls
Hull Operators and the Substitution Property
Filling in Dependencies
Algebraic and Topological Closures
Closures That Are Both Topological and Algebraic
Isomorphism and Cryptomorphy
Separation
Constructions on Closures
Small Closures


Alignments
Universal Algebras
Realizing Alignments by a Single Operation
Universality of Groupoids
Universality of Napoleonic Closure
Copoints in Finitary Closures
Copoints and the Separation Axioms 53 and 54
Matroids and Antimatroids
Independence
More Examples of Matroids and Antimatroids
Order Convexity
Acyclic Semigroups
Articulated Sums


Background Review
Functions
Relations
Orders
Basic Graph Theory
Some More Algebraic Examples
Two Binary Operations


Set Theory
Cardinals
The Axiom of Choice
Ordinals
Countability