# Category:Metric Spaces

This category contains results about Metric Spaces.
Definitions specific to this category can be found in Definitions/Metric Spaces.

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

## Subcategories

This category has the following 44 subcategories, out of 44 total.