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I was reading about discrete metric spaces and got a bit confused as to what exactly is the correct definition.

The first one that I came across is that a metric space $(D, \rho)$ is discrete if the accompanying metric is discrete, meaning $\rho(x,y) = 1$ if $x \neq y$ or $\rho(x,y)=0$ if $x=y$. If I understand that correctly then also space of continuous functions $C[0,1]$ equipped with such metric is discrete?

The second definition stated that a metric space $(D, \rho)$ is discrete if there exists some $\kappa > 0$ such that $\rho(x,y)\geq \kappa$ for every $x, y \in D$. By this definition then also $(\mathbb{N}, |\cdot|)$ is a discrete metric space even though the metric here is the absolute value of difference.

Thanks for any clarifications in advance!

Metod Jazbec
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  • Language is made as you go. Even though the definitions disagree, when you look at the topology defined by the metric, you get the same topology. Also, you can always scale (multiply by a positive function) a discrete metric from the second definition, to get the metric from the first one. By the way, the condition in the second definition should have $x\neq y$. –  May 03 '18 at 19:51
  • I see you have received a good answer... With the usual metric $d(x,y)=|x-y|$ for real numbers $x,y,$ the space $S={1/n:n\in \Bbb N}$ is an example of a discrete metric space even though for every $r>0$ there are $x,y\in S$ with $0<d(x,y)<r.$ – DanielWainfleet May 03 '18 at 21:26

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You are confusing the discrete metric with a discrete metric space.

Discrete metric

The discrete metric $\rho:X\times X\rightarrow [0,\infty)$ over a set $X$ is defined: $$\rho=\left\{\begin{matrix}0,&\text{ if x = y}\\1,&\text{ otherwise}\end{matrix}\right.$$

Discrete space

Let $(Y,d)$ be a metric space. $(Y,d)$ is called a discrete space if each $x\in Y$ is an isolated point. In other words, there exists a $\delta>0$ such that for every $y\in Y$ distinct from $x$ we have $d(x,y)>\delta$.

It is also clear that the discrete metric on a nonempty set constitutes a discrete space. I hope that clears up your confusion.

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Tom
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