Standard Discrete Metric is Metric
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Theorem
The standard discrete metric is a metric.
Proof
Let $d: S \times S \to \R$ denote the standard discrete metric on the underlying set $S$ of some space $\struct {S, d}$.
By definition:
- $\forall x, y \in S: \map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Proof of Metric Space Axiom $(\text M 1)$
\(\ds \map d {x, x}\) | \(=\) | \(\ds 0\) | Definition of Standard Discrete Metric |
So Metric Space Axiom $(\text M 1)$ holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality
Let $x = z$.
\(\ds \map d {x, z}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, y} + \map d {y, z}\) | \(\ge\) | \(\ds \map d {x, z}\) | Definition of Standard Discrete Metric |
Let $x \ne z$.
Either $x \ne y$ or $y \ne z$, or both.
So:
\(\ds \map d {x, y} + \map d {y, z}\) | \(\ge\) | \(\ds 1\) | Definition of Standard Discrete Metric | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \map d {x, z}\) | Definition of Standard Discrete Metric |
So in either case:
- $\map d {x, y} + \map d {y, z} \ge \map d {x, z}$
and Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 3)$
Let $x \ne y$.
\(\ds \map d {x, y}\) | \(=\) | \(\ds 1\) | Definition of Standard Discrete Metric | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {y, x}\) | Definition of $d$ |
So Metric Space Axiom $(\text M 3)$ holds for $d$.
$\Box$
Proof of Metric Space Axiom $(\text M 4)$
\(\ds x\) | \(\ne\) | \(\ds y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {x, y}\) | \(>\) | \(\ds 0\) | Definition of Standard Discrete Metric |
So Metric Space Axiom $(\text M 4)$ holds for $d$.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: The Definition
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $7$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.2$