:PROPERTIES:
:ID:       6f24f731-60e5-4904-88d7-c63869505981
:ROAM_ALIASES: metric
:END:
#+title: metric space
#+author: Preston Pan
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* Introduction
A metric space $(G, d)$ is a set with a metric $d(x,y): G \times G \rightarrow \mathbb{R}$ defined on members of the set.
This metric is a generalization of distance, with the following properties:
\begin{align}
\label{}
d(x, x) = 0 \\
x \ne y \implies d(x, y) > 0 \\
d(x, y) = d(y, x) \\
d(x, z) \le d(x, y) + d(x, z)
\end{align}
where property $(4)$ is the triangle inequality.