# Metric space

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$. The metric $d$ represents a distance function between pairs of points of $S$ which has the following properties:

• Symmetry: for all $x, y \in S$, $d(x, y) = d(y, x)$
• Non-negativity: for all $x, y \in S$, $d(x, y) \geq 0$
• Uniqueness: for all $x, y \in S$, $d(x, y) = 0$ if and only if $x = y$
• The Triangle Inequality: for all points $x, y, z \in S$, $d(x, y) + d(y, z) \geq d(x, z)$

Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

## Popular metrics

• The Euclidean metric on $\mathbb{R}^n$, with the "usual" meaning of distance
• The Discrete metric on any set, where $d(x,y)=1$ if and only if $x\neq y$
• The Taxicab metric on $\mathbb{R}^2$, with $d(((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|$