Topologies generated by relations

Abstract

Let R be a relation on a set X, and if A ⊂ X set RA = {x ∣ (x, α) ∈ R for some α ∈ A} and AR = {x ∣ (α, x) ∈ R for some α ∈ A}. Also A is called an antiset in case no two distinct elements of A are related. If A is a collection of antisets, then we generate a topology T(A) by taking sets of the form RA or AR (or X or ø) as subbasic open sets. Then conditions are given under which this topology satisfies separation axioms, or is compact or connected. For example. Theorem: Let A contain the singletons. If for each x ∈ X and y ∈ X \ x, there is a z ∈ X such that (x, z) ∈ R ((z, x) ∈. R) and (y, z) ∉ R ((z, y) ∉ R), then T(A) is a T₁-topology. The conditions used to obtain compactness or connectedness are analogous to the conditions used to get the same properties for the order topology on a totally ordered set. Finally, by modifying the definition of T(A) slightly, we obtain conditions so that if X is a tree and R the cutpoint order, then T(A) is the original topology.