Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1980-0565335-8
MathSciNet review: 565335
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}$.


References [Enhancements On Off] (What's this?)

  • [1] O. Frink, Topology in lattices, Trans. Amer. Math. Soc. 51 (1942), 569-582. MR 0006496 (3:313b)
  • [2] S. A. Kogan, Solution of three problems in lattice theory, Uspehi Mat. Nauk 11 (1956), 185-190. MR 0078333 (17:1176f)
  • [3] M. Kolibiar, Bemerkungen über Intervalltopologie in halbgeordneten Mengen, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961), Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 252-253. MR 0147424 (26:4940)
  • [4] Y. Matsushima, Hausdorff interval topology on a partially ordered set, Proc. Amer. Math. Soc. 11 (1960), 233-235. MR 0111705 (22:2567)
  • [5] F. S. Northam, The interval topology of a lattice, Proc. Amer. Math. Soc. 4 (1953), 824-827. MR 0057534 (15:244a)
  • [6] L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, New York, 1970. MR 0266131 (42:1040)
  • [7] E. S. Wolk, Topologies on a partially ordered set, Proc. Amer. Math. Soc. 9 (1958), 524-529. MR 0096596 (20:3079)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D10, 06B30, 54F05

Retrieve articles in all journals with MSC: 54D10, 06B30, 54F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0565335-8
Keywords: Interval topology, separation axioms, regular, normal, finitely separable
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society