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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

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

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  • $\begingroup$ The default topology on $\Bbb Q$ is Euclidean, not discrete. $\endgroup$ – Brian M. Scott Jul 7 '13 at 21:50
  • $\begingroup$ Ah, that was quite a brainfart; thanks. $\endgroup$ – fuglede Jul 8 '13 at 9:45
  • $\begingroup$ @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. $\endgroup$ – Pete L. Clark Jul 8 '13 at 9:50
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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.
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