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metric space

Definition/Summary
A metric space (A, d) is a set A paired with a function [itex]d: A \times A \to \mathbb{R}[/itex], such that for any elements x, y, z in A:

[tex]\begin{align*}
d(x,y) & \geq 0 \\
d(x,y) & = 0 \text{ if and only if } x = y \\
d(x,y) & = d(y,x) \\
d(x,z) & \leq d(x,y) + d(y,z)
\end{align*}[/tex]

The function d is referred to as a metric or distance function.

Equations

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Recent forum threads on metric space
 
Breakdown
Mathematics
> Calculus/Analysis
>> General Analysis

See Also

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Extended explanation
Examples of metric spaces:

The real numbers equipped with the metric d(x,y) = |x-y| is a metric space.

For any nonempty set A, the discrete metric can be defined as d(x,y) = 0 if x=y and d(x,y) = 1 otherwise, for all x, y in A.

Commentary

hunt_mat @ 05:07 AM Jul16-10
The first condition of a metric space [ctex]d(x,y)\geq 0[/ctex] can be derived from the last three in the following manner:
[ctex]
2d(x,y)=d(x,y)+d(y,x)\geq d(x,x) =0
[/ctex]
So this shows that [ctex]d(x,y)\geq 0[/ctex].