习题 24, § 2

设$G$和$G'$是两个局部同构连通的群。 a) 令$f$为$G$的局部同构和$G'$，定义在$G$的单位元素的邻域$V$上。 令$H$为乘积$G \times G'$的子群，由点$(x, f(x))$的集合生成，其中$x \in V$。 考虑映射$x \to (x, f(x))$下包含在$V$中的$e$的邻域的像的$H$由……组成上的过滤器基。 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 $0 < \theta < 1$. 证明对于这个本地同构，$H$上定义的拓扑$\mathfrak{T}$与$G \times G'$上的乘积拓扑在$H$上诱发的拓扑不同。