Exercise 6, § 3

Let $G$ be a topological group which satisfies the following conditions: (LR_I) $G$ is Hausdorff and locally connected.

(LR_{II}) There exists a connected neighbourhood $U$ of the identity element $e$ of $G$ such that the complement of $e$ in $U$ is not connected. Show that, under these conditions, $G$ is locally isomorphic to $\mathbf{R}$. [Prove first that if $V$ is any connected neighbourhood of $e$ in $G$ contained in $U$, then $V \cap C\{e\}$ is not connected, that $e$ lies in the closure of every component of $V \cap C\{e\}$, and that $V \cap C\{e\}$ has exactly two components, by reasoning as in Exercise 4. Then take the connected neighbourhood $V$ sufficiently small and define a linear ordering on $V$ such that $\mathscr{C}_0(V)$ is coarser than the topology induced on $V$ by the topology of $G$, and such that Proposition 2 of § 2 applies to the set $E$ of elements $\geq e$ of $V$.]