Discrete metric

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 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. 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.