Closures of weakened analytic groups

Abstract:

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