Exercise 4, § 3

Let $G = \mathbf{GL}(2, \mathbf{R})$ be the multiplicative group of invertible square $2 \times 2$ matrices with real elements. For each integer $n > 0$, let $V_n$ be the set of matrices $X = \begin{pmatrix} x & y \\ z & t \end{pmatrix} \in G$ such that $|x - 1| \leq 1/n, |y| \leq 1/n, |z| \leq 1/n, |t - 1| \leq 1/n$. Show that the family of the $V_n$ is a fundamental system of neighbourhoods of the identity element for a topology compatible with the group structure of $G$. With respect to this topology, $G$ is locally compact and the left and right uniformities on $G$ are distinct. \*