Examples of disconnected metric spaces that cannot be isometrically embedded into a connected metric space

Question

Please give me examples of disconnected metric spaces that cannot be isometrically embedded into connected metric spaces.

I have just started to study connected metric spaces and I haven't found out a metric space $X$ which isn't a subspace from another metric space such that $X$ isn't connected.

Every counterexample that I've looked, has been subspaces from a bigger metric space, e.g. $[0, 1]\cup[2, 3]$ which fails to be an interval in $(\mathbb{R}, |\cdot|$).

It might be topological space that isn't connected but for metric spaces? is there any apart from discrete metric spaces?

Thanks in advance

Answer 1

If $X$ is any bounded metric space then $C(X)$, the space of real continuous functions on $X$ is a connected metric space under the sup norm and the map $x \to f_x$, where $f_x(y)=d(x,y)$, is an isometric embedding of $X$ into $C(X)$. Hence the example you are looking for does not exist.