Separation axioms for interval topologies

Abstract:

In Theorem 1 of this note, results of Kogan [2], Kolibiar [3], Matsushima [4] and Wolk [7] concerning interval topologies are presented under a common point of view, and further characterizations of the ${{\text {T}}_2}$ axiom are obtained. A sufficient order-theoretical condition for regularity of interval topologies is established in Theorem 2. In lattices, this condition turns out to be equivalent both to the ${{\text {T}}_2}$ and to the ${{\text {T}}_3}$ 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 ${{\text {T}}_2}$ but not ${{\text {T}}_3}$.