Definition:Metric Space

Definition
A metric space $M = \left({A, d}\right)$ is an ordered pair consisting of:
 * $(1): \quad$ a set $A \ne \varnothing$

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 $M = \left\{{A, d}\right\}$, 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 $M = \left[\!\left[{A, \rho}\right]\!\right]$ can also be found.

Also see

 * Pseudometric, which is the same as a metric but does not include the condition M4.


 * Quasimetric, which is the same as a metric but does not include the condition M3.

Relation to Norms

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


 * Norm Defines Metric, where it is shown that any norm can be used to define a metric $d \left({x, y}\right) = \left\| {x - y} \right\|$.