Discrete Metric¶

Definition¶

Consider the space $\mathbb{R}^n$ with its elements $\boldsymbol{x}$ and $\boldsymbol{y}$. We define a discrete metric $d_{\mathrm{disc}}$ as:

Although the example uses the Real Coordinate Space $\mathbb{R}^n$, the same formula works on any set. It satisfies the Metric and Metric Space axioms: in particular, if $x\ne z$$z$, at least one of $x\ne y$ and $y\ne z$ holds, giving the triangle inequality.

Interpretation¶

The discrete metric is the simplest possible distance: distinct points are always exactly $1$ apart, and a point is $0$ from itself. It treats every pair of different points as equally far, so it only records whether two points are equal.

Every line between two of the four points has length $1$, however the points are drawn. The red circle of radius $\tfrac{1}{2}$ around $\boldsymbol{x}$ contains no other point: under this metric no point has neighbours closer than $1$.

The discrete metric is mainly a test case: it satisfies every metric axiom, yet little of the behaviour of the Euclidean distance carries over. Since $d_{\mathrm{disc}}(\boldsymbol{x}_i,\boldsymbol{x})<1$ forces $\boldsymbol{x}_i=\boldsymbol{x}$$\mathbf{x}$, a sequence converges only if it is eventually constant. In the same way, $d_{\mathrm{disc}}(\boldsymbol{x}_j,\boldsymbol{x}_k)<\tfrac{1}{2}$ forces $\boldsymbol{x}_j=\boldsymbol{x}_k$${\mathbf{x}}_k$, so every Cauchy sequence is eventually constant and the space is complete.

The metric also shows that a metric need not come from a norm: a norm would give $\|2\boldsymbol{x}\|=2\|\boldsymbol{x}\|$$2\Vert \mathbf{x}\Vert$, while $d_{\mathrm{disc}}(2\boldsymbol{x},\boldsymbol{0})=d_{\mathrm{disc}}(\boldsymbol{x},\boldsymbol{0})=1$ for $\boldsymbol{x}\ne\boldsymbol{0}$$0$. Points that are very close in the Euclidean distance are at discrete distance $1$, so no bound $d_{\mathrm{disc}}\le b\,d_2$ holds, which is why the discrete metric is not strongly equivalent to the $p$-metrics.

The discrete metric returns later in the course as a standard counterexample for open and closed sets.

Key Results¶

• The discrete metric is not strongly equivalent to any of the $p$-metrics.

Examples¶

• $d_{\mathrm{disc}}(2,5)=1$ since $2\neq 5$$5$.

Prerequisites¶

• Real Coordinate Space
• Metric and Metric Space

Related Concepts¶

• $p$-norm and $p$-metric
• Strongly Equivalent Metrics
• Convergence in a Metric Space

Sources¶

• Math III: Analysis · Lecture 4

Symbols: real numbers

Last updated: 20 October 2026 03:02