A topology generated by selections

Every selection f:F2(X)→X for the family F2(X) of at most two-point subsets of a set X naturally defines an order-like relation ⪯f on X by letting x⪯fy 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.