Closures of weakened analytic groups

Abstract

Let ( G , G ) (G,\mathcal {G}) be a topological group with dense subgroup L L , and suppose that L L is an analytic group in a topology τ \tau that is stronger than the topology that L L inherits from G \mathcal {G} . It is known that L L contains a τ \tau -closed abelian subgroup H H that completely determines the topology of L L . We now prove that the G \mathcal {G} -closure H ¯ \overline H of H H similarly determines the topology of G G . ( G , G ) (G,\mathcal {G}) always has a left-completion in the category of topological groups, and the properties of H ¯ \overline H determine whether ( G , G ) (G,\mathcal {G}) is locally compact, analytic, metrizable, left-complete, or finite dimensional. We discuss the relationship between these results and recent work of Goto.