On the Characterization of Hausdorff Spaces

In this paper A, B, X, Y denote sets. Let X be an empty set. Note that ⋃ X is empty. Next we state several propositions: (1) (δX) ◦A ⊆ [:A, A :]. (2) (δX) ([:A, A :]) ⊆ A. (3) For every subset A of X holds (δX) ([:A, A :]) = A. (4) dom〈π2(X×Y ), π1(X×Y )〉 = [:X, Y :] and rng〈π2(X×Y ), π1(X×Y )〉 = [:Y, X :]. (5) 〈π2(X × Y ), π1(X × Y )〉 ◦[:A, B :] ⊆ [:B, A :]. (6) For every subset A of X and for every subset B of Y holds 〈π2(X × Y ), π1(X × Y )〉 ◦[:A, B :] = [:B, A :]. (7) 〈π2(X × Y ), π1(X × Y )〉 is one-to-one. Let X, Y be sets. One can verify that 〈π2(X ×Y ), π1(X ×Y )〉 is one-to-one. The following proposition is true (8) 〈π2(X × Y ), π1(X × Y )〉 −1 = 〈π2(Y × X), π1(Y × X)〉.