Show that the discrete metric can not be obtained from $X\neq\{0\}$ If $X \neq \{ 0\}$ is a vector space. How does one go about showing that the discrete metric on $X$ cannot be obtained from any norm on $X$?
I know this is because $0$ does not lie in $X$, but I am having problems. Formalizing a proof for this.
This is also my final question for some time, after this I will reread the answers, and not stop until I can finally understand these strange spaces.

Say we have a normed field $K$ and that $X$ is a normed space over $K$.
The question has positive answer (i.e. discrete topology is not induced by any norm) if we implicitely assume that the normed field has a non-trivial norm. But if we further investigate into pathological cases, we can in fact find a normed space in which a trivial norm makes sense. So we have a counterexample.
Say $K$ is a finite field, then it can be shown that the only norm for $K$ is the trivial one. Now if we take the $K$-vector space $X = K^n$, then by fixing a basis (say, the canonical basis), there is a canonical way to define a norm over $X$, namely
$$
 |(x_1, \dots, x_n)| := \max_{i = 1, \dots, n} |x_i|,
$$
and it turns out to be a trivial norm over $X$. So the induced topology is the discrete topology.
So, to be fair, we can say that the trivial topology is not induced by any norm if we assume that $K$ has a non-trivial topology (the argument is the same given by Brian M. Scott and by user38268).
