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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Covering relations in the lattice of $ T\sb{1}$-topologies


Authors: Roland E. Larson and W. J. Thron
Journal: Trans. Amer. Math. Soc. 168 (1972), 101-111
MSC: Primary 54A10; Secondary 06A20
MathSciNet review: 0296883
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Abstract: A topology $ {\mathcal{T}_1}$ is said to cover another topology $ {\mathcal{T}_2}$ if $ {\mathcal{T}_2} \subsetneq {\mathcal{T}_1}$ and no other topology may be included between the two. In this paper, we characterize the relationship between a $ {T_1}$-topology and its covers. This characterization is used to prove that the lattice of $ {T_1}$-topologies is both upper and lower semimodular. We also prove that the sublattice generated by the covers of a $ {T_1}$-topology is isomorphic to the Boolean lattice of all subsets of the set of covers.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0296883-6
Keywords: Lattice of $ {T_1}$-topologies, covers, upper semimodular, lower semi-modular, Boolean lattice
Article copyright: © Copyright 1972 American Mathematical Society