Relationship between the meet and join operators in the lattice of group topologies

Abstract

Let L ( G ) L(G) be the lattice of all topologies on the group G G which make G G into a topological group. If τ 1 {\tau _1} and τ 2 {\tau _2} are Hausdorff group topologies and τ 1 ∨ τ 2 {\tau _1} \vee {\tau _2} is the discrete topology, then τ 1 ∧ τ 2 {\tau _1} \wedge {\tau _2} is a Hausdorff topology. If τ 1 {\tau _1} and τ 2 {\tau _2} are locally compact Hausdorff group topologies, then τ 1 ∨ τ 2 {\tau _1} \vee {\tau _2} is locally compact if and only if τ 1 ∧ τ 2 {\tau _1} \wedge {\tau _2} is Hausdorff.