Definition:Standard Discrete Metric

Definition

The standard discrete metric on a set $S$ is the metric satisfying:

$\map d {x, y} = \begin{cases} 0 & : x = y \\ 1 & : x \ne y \end{cases}$

This can be expressed using the Kronecker delta notation as:

$\map d {x, y} = 1 - \delta_{x y}$

The resulting metric space $M = \struct {S, d}$ is the standard discrete metric space on $S$.

Special Cases

Real Number Plane

The discrete metric on $\R^2$ is defined as:

$\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.

Also known as

This metric is also reported in some texts as the discrete metric.

Also see

• Standard Discrete Metric is Metric
• Definition:Discrete Space

• Results about the standard discrete metric can be found here.

Linguistic Note

Be careful with the word discrete.

A common homophone horror is to use the word discreet instead.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.

Sources

• 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.2$: Metric Spaces: Exercise $7$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{III}$: The Definition
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2.2$: Examples: Example $2.2.2$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $6$
• 1981: Karl R. Stromberg: An Introduction to Classical Real Analysis: $3.1$
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): $\S 1.1$: Open and Closed Sets: $(1.7)$