Exercise 5, § 1

Let $\mathfrak{B}$ be a filter on a group $G$, satisfying axioms $(\mathrm{GV}_1)$ and $(\mathrm{GV}_{\mathrm{II}})$.

a) There is a unique topology $\mathfrak{T}_s$ (resp. $\mathfrak{T}_d$) on $G$ such that, for each $a \in G$, the neighbourhood filter of $a$ with respect to $\mathfrak{T}_s$ (resp. $\mathfrak{T}_d$) is $a.\mathfrak{B}$ (resp. $\mathfrak{B}.a$).

Let $G_s$ (resp. $G_d$) be the topological space obtained by endowing $G$ with the topology $\mathfrak{T}_s$ (resp. $\mathfrak{T}_d$). For each $a \in G$, the left translation $x \to ax$ (resp. the right translation $x \to xa$) is a continuous mapping of $G_s$ into $G_s$ (resp. of $G_d$ into $G_d$). The symmetry $x \to x^{-1}$ is a continuous mapping of $G_s$ into $G_d$ and of $G_d$ into $G_s$.

b) The following conditions are equivalent: 1) $V$ satisfies $(\mathrm{GV}_{\mathrm{III}})$; 2) for each $a \in G$, $x \to xa$ is a continuous mapping of $G_s$ into $G_s$; 3) $x \to x^{-1}$ is a continuous mapping of $G_s$ into $G_s$.

*c) Let $G$ be the group $\mathbf{GL}(2, \mathbf{Q}_p)$ of $2 \times 2$ square matrices over the $p$-adic field $\mathbf{Q}_p$. For each integer $n > 0$, let $H_n$ be the set of

(*) If $G$ is locally compact, this condition implies that the topology of $G$ is compatible with its group structure (Chapter X, § 3, Exercise 25).

matrices of the form $I + p^n U$, where $U$ is a $2 \times 2$ matrix whose elements are $p$-adic integers. Show that the $H_n$ are subgroups of $G$ whose intersection consists only of the identity element. Deduce that if $\mathfrak{B}$ is the filter which has the $H_n$ as a base, then the corresponding topologies $T_s$ and $T_d$ on $G$ are Hausdorff. Show that these topologies are distinct.*