---
tag: 0NA4FK7B8S
kind: example
name: Discrete metric
msc: ["54E35"]
instance_of: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_space
    locator: Definitions
---

## Example

Let $X$ be a set.
For $x, y \in X$, let $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$.
Then $d$ is a [[metric|063W5JMY85]] on $X$, the discrete metric.

## Remarks

For each $x \in X$, the open ball of radius $1/2$ about $x$ is $\{x\}$. Thus each subset of $X$ is open, and the metric gives the [[discrete topology|0S2TEHX4E1]].
A sequence converges in this metric if and only if it is eventually constant.

## Explanations

### L1 human

In the discrete metric, two points are either equal or at distance $1$.
No point is close to another point, so the space has no interesting limits.