A topology generated by selections

Every selection f : F 2 ( X ) → X for the family F 2 ( X ) of at most two-point subsets of a set X naturally defines an order-like relation ⪯ f on X by letting x ⪯ f y if and only if f ( { x , y } ) = x . In particular, it also defines a natural interval-like topology on X. In the present paper we study properties of this topology, and its possible relations with other topologies on X with respect to which f is Vietoris continuous.