Metrics on $\mathbb R^{\mathbb N}$.

Solution 1:

This product topology is sequential, so you can simply work with sequences. Indeed, $\mathbb{R}^\mathbb{N}$ is a countable product of metric whence first countable spaces. So the product topology on $\mathbb{R}^\mathbb{N}$ is first countable. Hence we have a sequential space. That is, the topology is characterized by its converging sequences.

Recall that the product topology is also known as the topology of pointwise convergence. A sequence $x^{(n)}=(x_k^{(n)})$ converges to $x=(x_k)$ for the product topology if and only if $x_k^{(n)}\longrightarrow x_k$ in $(\mathbb{R},|\cdot|)$ for every $k$. Therefore:

Showing that the product topology is equal to the topology induced by $D$, is equivalent to proving: $$ D(x^{(n)},x)\rightarrow 0\quad\iff\quad |x^{(n)}_k-x_k|\rightarrow 0 \quad\forall k. $$

This holds whether you take $D(x,y)=\sup_k \frac{\min \{1,d(x_k,y_k)\}}{k}$ or $D(x,y)=\sum_k\frac{d(x_k,y_k)}{(1+d(x_k,y_k))2^k}$. Note that the exact same arguments apply more generally to $X^\mathbb{N}$ for any metric space $(X,d)$. The first $D$ is a bit easier. I'll do the second one. If you already know that $d'$ is topologically equivalent to $d$, you can jump to the last paragraph.

I assume you have checked that $$ D(x,y)=\sum_{k=0}^{+\infty}\frac{d'(x_k,y_k)}{2^k}\qquad\mbox{where }\;d'(s,t)=\frac{|s-t|}{1+|s-t|} $$ defines a metric on $\mathbb{R}^\mathbb{N}$. This essentially requires to check that $d'$ is a metric on $\mathbb{R}$.

Now observe that $d'$ is topologically equivalent to the standard distance $|s-t|$. Indeed, if $|s_n-s|\longrightarrow 0$, then clearly $d'(s_n,s)\longrightarrow 0$. Conversely, assume that $d'(s_n,s)\longrightarrow 0$. First, note that $|s_n-s|$ is bounded, for otherwise there would exist a susbequence such that $|s_{n_k}-s|\longrightarrow +\infty$, whence $d'(s_{n_k},s)\longrightarrow 1$. Second, let $M$ be an upper bound and note $d'(s_n,s)\geq \frac{|s_n-s|}{1+M}$. So $|s_n-s|\longrightarrow 0$.

Let $x^{(n)}=(x_k^{(n)})$ and $x=(x_k)$ in $\mathbb{R}^\mathbb{N}$. We need to show that $D(x^{(n)},x)\longrightarrow 0$ if and only if $x^{(n)}_k\longrightarrow x_k$ in $\mathbb{R}$ for every $k$ (with respect to the usual distance on $\mathbb{R}$). By the previous paragraph, the latter is equivalent to the condition: $d'(x^{(n)}_k,x_k)\longrightarrow 0$ for every $k$. So we need only show: $$ D(x^{(n)},x)\longrightarrow 0\qquad\iff\qquad d'(x^{(n)}_k,x_k)\longrightarrow 0\quad\forall k. $$

Forward direction: First, fix $k$ and observe that $d'(x^{(n)}_k,x_k)\leq 2^k D(x^{(n)},x)$ for every $n$. This direction follows immediately.

Backward direction: The series defining $D(x,y)$ converges normally on $\mathbb{R}^\mathbb{N}\times \mathbb{R}^\mathbb{N}$, as $\sum_{k\geq 0}\frac{1}{2^k}<\infty$. So we can swap sum and limits (dominated convergence for series). In particular $$ \lim_{n\rightarrow+\infty}D(x^{(n)},x)=\sum_{k\geq 0}\frac{1}{2^k}\lim_{n\rightarrow+\infty}d'(x^{(n)}_k,x_k)=0. $$