Exercise 24, § 2

Let $G$ and $G'$ be two locally isomorphic connected groups. a) Let $f$ be a local isomorphism of $G$ with $G'$, defined on a neighbourhood $V$ of the identity element of $G$. Let $H$ be the subgroup of the product $G \times G'$ generated by the set of points $(x, f(x))$ where $x \in V$. Consider the filter base on $H$ consisting of the images of neighbourhoods of $e$ contained in $V$ under the mapping $x \to (x, f(x))$. Show that this filter base is a fundamental system of neighbourhoods of the identity element of $H$ for a topology $\mathfrak{T}$ compatible with the group structure of $H$. b) Show that there exist two discrete normal subgroups $K, K'$ contained in the centre of $H$, such that $G$ is isomorphic to $H/K$ and $G'$ is isomorphic to $H/K'$ [use Exercise 5 b)]. c) Take $G$ and $G'$ each to be the group $T$; let $\varphi$ be the canonical homomorphism of $\mathbf{R}$ onto $T = \mathbf{R}/\mathbf{Z}$, and let $f$ be the local isomorphism which maps each point $\varphi(x)$, where $-1/4 \leq x \leq 1/4$, to $\varphi(\theta x)$, $\theta$ being an irrational number such that $0 < \theta < 1$. Show that for this local isomorphism, the topology $\mathfrak{T}$ defined on $H$ is distinct from the topology induced on $H$ by the product topology on $G \times G'$.