Are all metric space as a euclidean space?

Question

I believe that all euclidean space is a metric space?
But I need to know about inverse?
I mean: are all metric space as a euclidean space?
Is there any kind of metric space which is not euclidean space?
Show me sample!

Answer 1

Here are two very simple examples:

1. Let $X=\{0,1\}$, and define $d(0,0)=d(1,1)=0$ and $d(0,1)=d(1,0)=1$; then $\langle X,d\rangle$ is a metric space that is clearly not $\Bbb R^n$ for any $n$.

2. Let $X=\wp(\Bbb R)$, the set of all subsets of $\Bbb R$, and for $x,y\in X$ define $$d(x,y)=\begin{cases}0,&\text{if }x=y\\1,&\text{if }x\ne y\;.\end{cases}$$ Then $\langle X,d\rangle$ is a metric space that cannot even be embedded in any Euclidean space, because its cardinality is greater than that of $\Bbb R^n$ for any $n$.