Separation axioms for interval topologies

In Theorem 1 of this note, results of Kogan [2], Kolibiar [3], Matsushima [4] and Wolk [71 concerning interval topologies are presented under a common point of view, and further characterizations of the T2 axiom are obtained. A sufficient order-theoretical condition for regularity of interval topologies is established in Theorem 2. In lattices, this condition tuns out to be equivalent both to the T2 and to the T3 axiom. Hence, a Hausdorff interval topology of a lattice is already regular. However, an example of a poset is given where the interval topology is T2 but not T3.