Metric space

Definition

A metric space is a set $X$ along with a distance function $d:X\times X\to \mathbb{R}$ such that the following hold:

• $d(x,y)\ge 0$ (non-negativity)
• $d(x,x)=0⟺x=0$ (identity of indiscernibles)
• $d(x,y)=d(y,x)$ (symmetry)
• $d(x,y)+d(y,z)\ge d(x,z)$ (triangle inequality)

A distance function satisfying all the above three conditions is termed a metric.

Induced topology

There is a natural induced topology on any metric space: the topology whose basis is open balls of positive radii about points in the metric space. Here, by open ball of radius $r$ about $x$ we mean the set of points $y$ such that $d(x,y)<r$.

A topological space which arises via the induced topology on a metric space, is termed metrizable. There may be many different metrics yielding the same topology, for instance the taxicab metric and the Euclidean metric for Euclidean space.

Facts

The metric is a jointly continuous function from the metric space to $\mathbb{R}$. This follows from the various axioms for the function.