AbstractK. Steffens (“Injektive Auswahlfunktionen,” Schriften aus dem Gebiet der Angewandten Mathematik Nr. 2, Aachen, 1972) has shown that a family of finite sets has a transversal if and only if the collection of all “critical” subfamilies forms a topology. In this paper these “transversal topologies” are characterized, as well as families whose transversal topologies satisfy separation axioms