# 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

• Who says it is "so important"? – Chris Eagle Jul 7 '13 at 16:19
• THe discrete topology can offer (counter)examples in several cases. E.g. all sets are both open and closed. – Calvin Lin Jul 7 '13 at 16:22
• Well it's a great source of counter examples for one thing. – Dan Rust Jul 7 '13 at 16:22

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.

• The default topology on $\Bbb Q$ is Euclidean, not discrete. – Brian M. Scott Jul 7 '13 at 21:50
• Ah, that was quite a brainfart; thanks. – fuglede Jul 8 '13 at 9:45
• @Brian: not in arithmetic geometry. Much of the importance of the adele ring $\mathbb{A}^1_{\mathbb{Q}}$ en.wikipedia.org/wiki/Adele_ring is that it is a locally compact group in which $\mathbb{Q}$ sits inside as a lattice: i.e., a discrete subgroup with compact quotient. – Pete L. Clark Jul 8 '13 at 9:50

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:

1. is trivially continuous if $X$ has the discrete topology or $Y$ has the trivial topology.
2. must be constant in order to be continuous, if $X$ has the trivial topology and $Y$ has the discrete topology.