§1 Basics

We want to formalise our intuition about distances in the real world, and try to generalise.

1.1 Definitions and Examples

Defn. (metric space)

Let $X$ be any set.

A metric on $X$ is a function $d:X \times X \rightarrow \mathbb{R}$ such that:

• $d(x,y) \geq 0$, equality iff $x=y$ ("positive semi-definite")
• $d(x,y) = d(y,x)$ ("symmetric")
• $d(x,y) + d(y,z) \geq d(x,z)$ ("triangle inequality")

We say $(X,d)$ is a metric space.