Separation axioms for interval topologies
Marcel Erné · Proceedings of the American Mathematical Society · 1980
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 T 2 {{\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 T 2 {{\text {T}}_2} and to the T 3 {{\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 T 2 {{\text {T}}_2} but not T 3 {{\text {T}}_3} .