Bases and Refinements of Topologies

Let A be a set and let B be a non empty set. Observe that B is non empty. In this article we present several logical schemes. The scheme FraenkelInvolution deals with a non empty set A, subsets B, C of A, and a unary functor F yielding an element of A, and states that: B = {F(a); a ranges over elements of A : a ∈ C} provided the parameters have the following properties: • C = {F(a); a ranges over elements of A : a ∈ B}, and • For every element a of A holds F(F(a)) = a. The scheme FraenkelComplement1 deals with a non empty relational structure A, a family B of subsets of A, a set C, and a unary functor F yielding a subset of A, and states that: B = {−F(a); a ranges over elements of A : a ∈ C} provided the parameters meet the following requirement: • B = {F(a); a ranges over elements of A : a ∈ C}. The scheme FraenkelComplement2 deals with a non empty relational structure A, a family B of subsets of A, a set C, and a unary functor F yielding a subset of A, and states that: B = {F(a); a ranges over elements of A : a ∈ C}