# Axiom:Metric Space Axioms

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## 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:

\((\text M 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \map d {x, x} = 0 \) | ||||||

\((\text M 2)\) | $:$ | \(\ds \forall x, y, z \in A:\) | \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | ||||||

\((\text M 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds \map d {x, y} = \map d {y, x} \) | ||||||

\((\text M 4)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds x \ne y \implies \map d {x, y} > 0 \) |

## Also defined as

The numbering of the 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

## 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