Definition:Metric Space

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Definition

A metric space $M = \left({A, d}\right)$ 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:
\((M1)\)   $:$     \(\displaystyle \forall x \in A:\) \(\displaystyle d \left({x, x}\right) = 0 \)             
\((M2)\)   $:$     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \)             
\((M3)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle d \left({x, y}\right) = d \left({y, x}\right) \)             
\((M4)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \)             


Points of Metric Space

The elements of $A$ are called the points of the space.


Distance Function

The mapping $d: A \times A \to \R$ is referred to as a distance function on $A$ or simply distance.


Triangle Inequality

Axiom $M2$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.


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.
  • Results about metric spaces can be found here.


In Relation to Norms

$d \left({x, y}\right) = \left\Vert {x - y} \right\Vert$


In Relation to Topological Spaces


Sources