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I am learning about metric spaces on my own. I have read the definition of discrete metric. But I want to understand it more clearly.

My question is as follows: Let $X = \{x_1,x_2, \cdots x_n\}$ be a set. What happens if we endow the set by the discrete metric?

Or, Consider $\mathbb{R}$ the set of real numbers. What happens if we consider $\mathbb{R}$ with the discrete metric?

One thing I understood that every subset of $X$ (or $\mathbb{R}$) is open and closed. But I want to know about the nature of the space.

If we consider $\mathbb{R}$ with the discrete metric, then every pair of distinct point of $\mathbb{r}$ has distance $1$. But take $2, 5$, these two points are distinct, but there distance not equal to $1$.

Please correct me.

Gayn Das
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    Welcome to MSE. If the distance from $2$ to $5$ is not $1$, that what is the value of that distance? – José Carlos Santos Jun 26 '22 at 19:49
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    When you speak about "distance" you should always specify the metric you use to measure it. – Surb Jun 26 '22 at 19:49
  • Note that the discrete metric and the usual metric on $\mathbb R$ are not the same thing. One and the same set can be equipped with completely different metrics, which completely changes what "distance" means. What the "distance" between two points is depends entirely on the metric. With the usual metric, the distance between 2 and 5 may not be 1. But with the discrete metric it is, by definition of the discrete metric. – Vercassivelaunos Jun 26 '22 at 20:04

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I think there's a need to explain the basics. To define the metric space one need to define the set of points and the metric, which is a function, which says what is a distance between any two points. The idea is that we can consider may metrics on one set and with respect to each of them the distance between points can be different.

For example the distance between $2$ and $5$ is $3$ with respect to euclidean distance but with respect to discrete metrics the distance is $1$. Each time you read about the distance you have to be certain which metric is considered.

Discerete distance works this way: any two different points are far away (distance is one). This works like paying for the phone. If you want to talk to somebody in your home, you pay nothing, since you don't need to use a phone. If you want to talk to your friend that is in a different building, you call him and it doesn't matter if he is your neighbour or he lives five streets away.

Back to mathematics, on the real line $\Bbb R$ with the discrete metric all the different points are equally distant. So going from $0$ to $1$ 'costs the same' as going from $0$ to $100$. That's it. Observe that from the very definition of the metric, $d(x,y)$ must be zero if $x=y$, so this part of the definition of the discrete metric is mandatory.

Why to consider discrete metric

You can consider many different metrics or topologies on a particular set. The choice is based not on how easy is to work with them but what are their properties (and what do you need).

  • Discrete metric is simple to define and work with, so it's presented very early in textbooks and lectures. It introduces beginners into the world of topology.
  • Discrete topology is extreme (the finest), so it's important to understand the properties of it in order to understand the others and to understand relations between them. It's important to understand that the finer the topology is the 'closer' to discrete topology. Since it's extreme, it has special properties, like it's the only topology that each function defined on this space is continuous.
  • Discrete topology is important to understand, to know when it's compact, when it's connected, when discrete metric is complete and so on, because we come across it many times working with other metrics or topologies. For example, some subsets of some spacec turn out to be discrete.
  • Discrete metric is often used as a source of counterexamples.
  • However discrete metric is in many cases too strong and in some sense boring.
Mateo
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  • Thanks for your answer. But I don't have enough reputations to upvote your answer. – Gayn Das Jun 27 '22 at 06:37
  • I have another question related to discrete metric. Is there anything special so that we endow a set with a discrete metric? Like I am saying that if I endow a set by a discrete metric, then we can easily see all its (of the space) topological property (e.g. open sets, closed sets etc.), while it is tough to say about is topological property when endowing with other metrics. Am I correct? – Gayn Das Jun 27 '22 at 06:42
  • I added a part 'Why to consider discrete metric'. – Mateo Jun 27 '22 at 07:35