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A chapter on topology talks about this and asks me to "think of a model for the discrete metric on $M$" where $M$ may have $1, 2, 3, 4$ points. Here is what I think:

$2$ points = $2$ distinct points

$3$ points = equilateral triangle

$4$ points = tetrahedron

$1$ point = ??? A point by itself? But that would mean $1 = 0$...

The book goes on to say "imagine the discrete metric on $\mathbb{R}$", which I'm unable to grasp. Thanks for your help!

JKnecht
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Math Damon
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    Imagining the discrete metric on $\mathbb R$ is a good way to make your head explode. You require an uncountable number of points an equal length apart from each other. Even trying to imagine $\mathbb N$ or $\mathbb Q$ under the discrete metric becomes difficult.

    I should also ask: what would the discrete topology on a set of 5 points look like?

     – Ian Coley Mar 16 '13 at 19:16
  • There is nothing wrong with one point by itself. – Karolis Juodelė Mar 16 '13 at 19:17
  • Wouldn't this discrete topology be the 'complete graph' ? – amr Mar 16 '13 at 19:36
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    Just imagine a continuous-dimensional simplex. – Loki Clock Mar 16 '13 at 20:36

2 Answers2

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I am not sure, why do you think about objects from continuous geometry, the following example shall work I guess. A finite graph is a finite set $V$ of points, and $E\subseteq V\times V$ are edges of this graph. We can endow $V$ with a metric by $$ d(x,y) = n\in \Bbb N_0\quad \Leftrightarrow \quad \text{any path from }x\text{ to }y\text{ passes at least }n\text{ edges} $$

The graph is called fully connected if $E = V\times V$. Then $d$ brings a discrete topology on a fully connected graph: from any point on such graph, you can reach any other point passing by only $1$ edge. If there is only one point, of course you don't have to move anywhere.

SBF
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  • I see this is a very old post. But I really liked this kind of concept. I would really love to know more about this. Can you please tell me more about this field, or simply suggest me a reference or something. – user398623 Sep 25 '17 at 14:37
  • @user398623 graph theory? – SBF Sep 25 '17 at 14:41
  • Graph theory? Really? And exactly which part of graph theory it is? Is Continuos Geometry as you mentioned a part of Graph Theory? – user398623 Sep 26 '17 at 03:45
  • @user398623: which exactly concept did you like? – SBF Sep 26 '17 at 09:19
  • I couldn't visualize the concept fully. That's why I was asking for some reference. Exactly when you said that we could reach any other passing by only 1 edge. – user398623 Oct 18 '17 at 15:43
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Hint: Stop imagining your topologies as belonging to Euclidean space. If you were going to imagine a discrete metric on $\mathbb{R}$ by visualizing the placement in $\mathbb{R}^3 $, there simply wouldn't be any room!

In the case of $ \mathbb{R} $, imposing the discrete metric just results in a case where every point $ x \neq y $ is of distance 1 away from each other. It just refers to the points being "separated", you're not meant to visualize the points separated in Euclidean space.

liamdalton
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