Category:Definitions/Examples of Metric Spaces

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This category contains definitions of examples of Metric Space.

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