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Transactions of the American Mathematical Society

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



Covering relations in the lattice of $T_{1}$-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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Keywords: Lattice of <IMG WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${T_1}$">-topologies, covers, upper semimodular, lower semi-modular, Boolean lattice
Article copyright: © Copyright 1972 American Mathematical Society