Does the notion of dimensions make sense in discrete metric space? For the metric space defined by distance formula,
$$d(p,q) = \begin{cases}1,&\text{if }p \ne q\\
0,&\text{if }p  = q\;,\end{cases}$$
does the notion of number of dimensions exist?
If yes, then is dimension of the above metric space uncountable?
Can a discrete metric space exist in finite dimensions?
 A: There are several different notions of dimension for topological spaces in general and hence metric spaces in particular, including small and large inductive dimension and covering dimension; all discrete metric spaces have dimension $0$ in all of these senses. There is also a notion of Hausdorff dimension, that applies specifically to metric spaces. If your discrete metric space is countable, its Hausdorff dimension is also $0$; if it’s uncountable, its Hausdorff dimension is $\infty$. There are other notions of dimension for metric spaces, but these are the most familiar.
A: As a manifold, this object is zero dimensional, as every point is open, the set of all points is an open cover, and the unique map $\{p\}\to\mathbb{R}^0=\{0\}$ is a homeomorphism for each $p$.
(Edited after Ilya's comment: This assumes the metric space is countable, otherwise it is not second countable and thus not a manifold.)
There are also other notions of dimension for metric spaces though, such as Hausdorff dimension, and they may not agree with this one. For a start many metric spaces are not manifolds, so this approach won't give you an answer at all.
