Remark

(1) Theorem 1 and its corollaries are no longer true if $\mathbf{R}$ (resp. $\mathbf{Q}_p$) is replaced by $\mathbf{C}$ (Exercise 1).

(2) Let G be a topological group. It can be shown† that the following conditions are equivalent: (a) G is a finite-dimensional real Lie group; (b) G is locally compact and there exists a neighbourhood of $e$ containing no subgroup distinct from $\{e\}$; (c) there exists an open neighbourhood of $e$ homeomorphic to an open ball of a space $\mathbf{R}^n$. (For a much less difficult result, cf. Exercise 6.)