:PROPERTIES:
:ID:       6f24f731-60e5-4904-88d7-c63869505981
:ROAM_ALIASES: metric
:END:
#+title: Metric Space
#+author: Preston Pan
#+description: The basis of modern analysis.
#+options: broken-links:t

* Introduction
A metric space $(X, d)$ is a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]] with a metric $d(x,y): X \times X \rightarrow \mathbb{R}$ 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. Also, the metric generates the [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] on the open sets; a basis can be chosen by including every
open ball, which is defined as $B(x, r) = \lbrace  y: d(x, y) < r\rbrace$. A neighbourhood basis can be chosen by including every open rational ball
that is a neighbourhood of $x$, and in fact this neighbourhood basis is countable, so metric spaces are first countable.