Algebraic obstructions to sequential convergence in Hausdorff abelian groups.
Let A be a locally convex, unital topological algebra whose group of units is open and such that inversion is continuous. Then inversion is analytic, and thus is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group is an analytic Lie group without...
We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.