Separation axioms for interval topologies

Abstract

In Theorem 1 of this note, results of Kogan [ <bold>2</bold> ], Kolibiar [ <bold>3</bold> ], Matsushima [ <bold>4</bold> ] and Wolk [ <bold>7</bold> ] concerning interval topologies are presented under a common point of view, and further characterizations of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {T}}_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {T}}_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript 3"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {T}}_3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {T}}_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but not <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript 3"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {T}}_3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .