Can any set be turned into a metric space? Let $X$ be any set, let $x,y\in X$ define a metric $d:X\times X\rightarrow \mathbb{R}$ by:
$$
d(x,y) = \left\{\begin{array}{cc}
1 & x\neq y \\
0 & x= y
\end{array}\right.
$$
Obviously $d$ is a valid metric.
If any set can become a metric space why are they so special? Is the only interesting part of a metric space the distance function?
EDIT
Some context:
I was trying to prove a result about metric spaces namely,
Theorem: If $X$ is a metric space with distance function $d:X\times X\rightarrow \Bbb{R}$ and $\varphi:X\rightarrow Y$ is a bijection then $Y$ is metrisable with a distance function $d'(y_1,y_2) = d(\varphi^{-1}(y_1),\varphi^{-1}(y_2))$
I remember feeling all accomplished after proving this result, but it seems rather trivial in light of this. I later found out that this is an actual result presented in some textbooks. Is there any real use for this theorem?
Best Answer
The metric spaces are special because the metric is special. Your question is really vague. So I cannot add much more than that. I, however, mention that the Euclidean space $\mathbf{R}^d$ is not the same metric space (not even the same topological space) as the space $\mathrm{X}$ whose "underlying set" is $\mathbf{R}^d$ and whose metric is the one you give.
ANSWER TO EDIT: The result is useful because it allows you to transport, so to speak, a metric (which you regard as given and special) from the metric space $\mathrm{X}$ onto the set $\mathrm{Y}.$