Definition 1.1

Definition 1.1: A metric space is a pair $$(X,d)$$ where $$X$$ is a set and $$d: X \times X \rightarrow \mathbb{R}$$ is a function with the following properties:

1. $$d(x,y) \geq 0$$ for any $$x,y \in X$$; $$d(x,y)=0$$ if and only if $$x=y$$

2. $$d(x,y) = d(y,x)$$ for all $$x,y \in X$$

3. $$d(x,y) + d(y,z) \geq d(x,z)$$ for all $$x,y,z \in X$$

We will often say "$$X$$ is a metric space" if the metric $$d$$ is understood from context.