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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Separation axioms for interval topologies


Author: Marcel Erné
Journal: Proc. Amer. Math. Soc. 79 (1980), 185-190
MSC: Primary 54D10; Secondary 06B30, 54F05
MathSciNet review: 565335
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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}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0565335-8
PII: S 0002-9939(1980)0565335-8
Keywords: Interval topology, separation axioms, regular, normal, finitely separable
Article copyright: © Copyright 1980 American Mathematical Society