# metric space

## Table of Contents

## 1. 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.