Why is the discrete metric said to be so important Can anyone enlighten me as to why the discrete metric is considered to be important in mathematics? The only real use I can see of it is that it shows the existence of a metric on any non-empty set.
I wonder if there is something I'm missing: maybe it used as a technique to prove certain types of theorems, or as a construction for other quantities (maybe characteristic functions) or in some particular application.
Thanks,
Matt
 A: I agree with the comments that it is always good to have that metric/topology in mind when first coming up with examples. On the other hand, a lot of spaces (think e.g. $\mathbb{N}$, $\mathbb{Z}$) are discrete in nature.
As far as using it to prove results, here's the first thing that came to my mind, which I remembered feeling was almost cheaty the first time I saw it:
If $X$ is a connected topological space and $f : X \to Y$ is a locally constant map from $X$ to any set $Y$, then $f$ is constant.
Proof. If we endow $Y$ with the discrete metric/topology, then $f$ is automatically continuous. It follows that $f(X)$ is connected, but connected components of $Y$ are points, so $f$ is constant.
A: In topology, both trivial topology and discrete topology (endowed by the discrete metric) are the most extreme examples of topology. A map $f:X\rightarrow Y$ between topological spaces:


*

*is trivially continuous if $X$ has the discrete topology or $Y$ has the trivial topology.

*must be constant in order to be continuous, if $X$ has the trivial topology and $Y$ has the discrete topology.

