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)

Induced topology on a metric space

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.