Covering relations in the lattice of 𝑇₁-topologies

Larson, Roland E.; Thron, W. J.

doi:10.1090/S0002-9947-1972-0296883-6

Covering relations in the lattice of $T_{1}$-topologies
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by Roland E. Larson and W. J. Thron PDF

Trans. Amer. Math. Soc. 168 (1972), 101-111 Request permission

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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