Definition:Metric Space

Definition
A metric space $M = \struct {A, d}$ is an ordered pair consisting of:
 * $(1): \quad$ a non-empty set $A$

together with:
 * $(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:

Notation
Some authors use the suboptimal $M = \set {A, d}$, which leaves it conceptually unclear as to which is the set and which the metric. This adds unnecessary complexity to the underlying axiomatic justification for the existence of the very object that is being defined.

The notation $M = \eqclass {A, \rho} {}$ can also be found.

Also see

 * Pseudometric, which is the same as a metric but does not include the condition $(\text M 4)$.


 * Quasimetric, which is the same as a metric but does not include the condition $(\text M 3)$.

In Relation to Norms

 * Metric Defines Norm iff it Preserves Linear Structure, where a homogeneous and translation invariant metric $d$ can be used to define a norm on a vector space.


 * Metric Induced by Norm is Metric, where it is shown that any vector space norm can be used to define a metric:
 * $\map d {x, y} = \norm {x - y}$


 * Metric Induced by Norm on Normed Division Ring is Metric, where it is shown that any division ring norm can be used to define a metric:
 * $\map d {x, y} = \norm {x - y}$

In Relation to Topological Spaces

 * For a metric space $\struct {A, d}$, one can define the topology $\tau$ on $A$ induced by the metric $d$, thus making $\struct {A, \tau}$ a topological space. Thus topological notions carry over to metric spaces.