Compatible group topologies

Abstract:

Two topologies defined on some space are compatible if they contain in common a Hausdorff topology. The following result is proved for two compatible group topologies ${\mathcal {A}_1}$ and ${\mathcal {A}_{_2}}$. Suppose ${\mathcal {A}_1}$ is locally compact and ${\mathcal {A}_2}$ is locally countably compact, and there is a non-void ${\mathcal {A}_2}$-open set contained in some ${\mathcal {A}_1}$-Lindelöf set. Then ${\mathcal {A}_1} \subseteq {\mathcal {A}_2}$. This result is a stronger version of a theorem by Kasuga, in which two group topologies are shown to be equal if both of them are locally compact and $\sigma$-compact, and they are compatible.