Abstract
This chapter discusses the concept of applications. It discusses the axiom of choice that is applied to concrete situations in certain parts of mathematics. It also discusses filters. Filters form very common and convenient tools in many parts of mathematics, for example, in topology, analysis, and model theory. One can define filters on partially ordered sets, for example, lattices or Boolean algebras. The chapter discusses filters of sets without reference to topological or algebraic structure. It has been observed that the subsets of a fixed set V behave under the operations of intersection, union, and complementation in a way strongly reminiscent of elementary algebra. The chapter discusses structures with relations and structures with operations. In mathematical practice, one usually deals with structures with operations, such as groups, rings, fields, and vector spaces. It is therefore convenient to extend the concept of structure so as to cover structures with operations.
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