Axiom:Metric Space Axioms

Definition

Let $A$ be a set upon which a distance function $d: A \times A \to \R$ is imposed.

The metric space axioms are the conditions on $d$ which are satisfied for all elements of $A$ in order for $\struct {A, d}$ to be a metric space:

\((\text M 1)\)	$:$			\(\ds \forall x \in A:\)	\(\ds \map d {x, x} = 0 \)
\((\text M 2)\)	$:$		Triangle Inequality:	\(\ds \forall x, y, z \in A:\)	\(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)
\((\text M 3)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds \map d {x, y} = \map d {y, x} \)
\((\text M 4)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds x \ne y \implies \map d {x, y} > 0 \)

Also defined as

The numbering of the metric space axioms is arbitrary and varies between authors.

It is therefore a common practice, when referring to an individual axiom by number, to describe it briefly at the same time.

Some sources replace $(\text M 1)$ and $(\text M 4)$ with a combined axiom:

\((\text M 1')\)

$:$

\(\ds \map d {x, y} \ge 0; \quad \forall x, y \in A:\)

\(\ds \map d {x, y} = 0 \iff x = y \)

thus allowing for there to be just three metric space axioms.

Others use:

\((\text M 1')\)

$:$

\(\ds \quad \forall x, y \in A:\)

\(\ds \map d {x, y} = 0 \iff x = y \)

as the stipulation that $\map d {x, y} \ge 0$ can in fact be derived.

Also see

• Distance Function for Distinct Elements in Metric Space is Strictly Positive

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): One: Metric Spaces: $1$: Open and Closed Sets