I've got a lot of problems imagining how open balls look like in metric spaces. This prevents me getting better insight in some proofs and exercises.
An example is the $d_1$-metric defined as follows: $$d_1(x,y) = \frac{|x-y|}{1+|x-y|} (x,y \in \mathbb{R})$$ and the $d_2$-metric on a function space $C([0,1])$: $$d_{\infty}(f,g) = sup\{|f(t) - g(t)| | t \in [0,1]\} \ \ \ (f,g \in C([0,1]).$$ For the $d_1$-metric an open ball centered at $x$ and radius $r>0$ is $$ B_1(x,r) = \begin{cases}\mathbb{R}&\text{if $r \geq1$}\\ \left]x - \frac{r}{1-r} , x + \frac{r}{1-r} \right[ \hspace{10mm}\text{if $r < 1$}\end{cases}$$ I don't quite see how the open ball for $r \geq 1$ can be $\mathbb{R}$.
For the $d_{\infty}$-metric I have an idea on how to draw it in two dimensions, but I'm not sure. Can anyone enlighten this for me (maybe in 3 dimensions)? Also, how do you imagine it? Is there some simple trick to 'see' it, maybe by looking at some critical points?
Note that $$d_1(x,y) = \frac{|x-y|}{1+|x-y|}<1 (x,y \in \mathbb{R})$$
Therefore if $r>1$ every point $y\in \mathbb {R}$ satisfies $$d_1(x,y)<r$$
Thus the open ball centered at $x$ with radius $r$ contains all of the real line.
The open balls for $d_\infty $ metric are ribbons or tubes around the central function depending on the dimension.