Exercise 15, § 1

General Topology, VII

It is set in small type as supplementary.

Let $I$ be a closed cube in $\mathbf{R}^n$ with side $2\pi$ . Map each point $x = (x_i)$ of $I$ to the point $y = (y_j)$ of $\mathbf{R}^{n+1}$ such that

\begin{align*} y_1 &= \sin x_1 \\ y_2 &= (2 + \cos x_1) \sin x_2 \\ &\cdots \\ y_p &= (2^{p-1} + \sum_{k=1}^{p-1} 2^{p-k-1} \cos x_{p-k} \cos x_{p-k+1} \cdots \cos x_{p-1}) \sin x_p \quad (2 \leq p \leq n) \\ &\cdots \\ y_{n+1} &= \sum_{k=1}^{n-1} 2^{p-k-1} \cos x_{n-k} \cos x_{n-k+1} \cdots \cos x_n. \end{align*}

Show that the image of I under this mapping is homeomorphic to $\mathbf{T}^n$ (proof by induction on $n$ : observe that $r_p$ has the sign of $\sin x_p$ for $\leq p\ n$ ). If $n = 2$ , the subset of $\mathbf{R}^3$ so defined is called a torus of revolution.

(16) Let G be a subgroup of $\mathbf{R}^n$ which contains a compact connected subset K such that the affine linear variety generated by K is of dimension $p$ . Show that G contains a vector subspace of dimension $p$ . [Proof by induction on $n$ , using Exercise 10 c) of Chapter VI, § 1.]

(17) Let E be the product $(\mathbf{Q}_p)^n$ of $n$ factors equal to the field $\mathbf{Q}_p$ of $p$ -adic numbers (Chapter III. § 6, Exercise 23), endowed with the topological group structure which is the product of the additive topological structures of the factors $\mathbf{Q}_p$ , and with its $n$ -dimensional vector space structure over the field $\mathbf{Q}_p$ . If $v$ denotes the additive $p$ -adic valuation on $\mathbf{Q}_p$ , put $x_p = p^{-v_p x}$ for all $x \neq 0$ in $\mathbf{Q}_p$ , and $o_p = 0$ ; and put $\|x\| = \sup x_i$ for all $x = (x_i)_{1 \leq i \leq n} \in E$ (cf. Chapter IX. § 3, nos. 2 and 3).

a) A subset A of E is relatively compact if and only if

\sup_{x \in A} \|x\| < +\infty.

[Note that

t_p

is a continuous mapping of

\mathbf{Q}_p

into

\mathbf{Z}

(with the discrete topology).]

b) If G is a closed subgroup of E, then G is a topological module over the ring $\mathbf{Z}_p$ of $p$ -adic integers; if $x \in G$ , we have $nx \in G$ for all $n \in \mathbf{Z}$ , and $\mathbf{Z}$ is dense in $\mathbf{Z}_p$ .

c) If K is a compact subgroup of E, there exists a free system of $m \leq n$ points $a_i$ ( $1 \leq i \leq m$ ) of E, such that K is the direct sum of the m groups $\mathbf{Z}_p \cdot a_i$ . [Use a) to show that, if $(e_j)_{1 \leq j \leq n}$ is the canonical base of E, there is an integer $k \in \mathbf{Z}$ such that K is contained in the direct sum of the n groups $\mathbf{Z}_p \cdot p^k e_j$ ; then apply the theory of modules over a principal ideal ring.]

d) If G is a closed (but not compact) subgroup of E, then G contains a vector subspace of dimension 1, $\mathbf{Q}_p \cdot a$ with $a \neq 0$ . [Let C be the compact subset of E consisting of all points $x \in E$ such that $\|x\| = 1$ ; show that $G \cap C$ contains a sequence of points $(x_r)_{r \in \mathbf{N}}$ such that $p^{-r} x_r \in G$ , and take $a$ to be a cluster point of this sequence.]

e) Let G be a closed subgroup of E, let V be the largest vector subspace contained in G and let W be a vector subspace complementary to V; show that $W \cap G$ is compact and that G is the direct sum of V and $W \cap G$ (argue as in Theorem 2 of no. 2).

f) Given a subgroup G of E, let $G^*$ denote the set of all $u = (u_i) \in E$ such that

(u|x) = \sum_{i=1}^n u_i x_i

belongs to $\mathbf{Z}_p$ for all $x = (x_i) \in G$ . Show that $G^*$ is closed in E, that $(\overline{G})^* = G^*$ and that $(G^*)^* = \overline{G}$ .

g) State and prove the analogues of Exercises 2 to 8 inclusive for closed subgroups of E. In particular, every closed subgroup and every Hausdorff quotient group of a group of the form

(\mathbf{Q}_p)^r \times (\mathbf{Q}_p/\mathbf{Z}_p)^s \times (\mathbf{Z}_p)^t \times F

(where F is the product of a finite number of cyclic groups of $p$ -power order) is a group of the same form.

Solution

No solution has been written yet.