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

Further information: Metric induces 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 $d(x,y)<r$. The fact that this works requires the use of the axioms of symmetry and triangle inequality.

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. For full proof, refer: Metric is jointly continuous
• The map that sends a metric space to its associated topological space, is a functor. In other words, a continuous map between metric spaces is a continuous map of the underlying topological spaces. For full proof, refer: Induced topology from metric is functorial
• The topology arising from the metric induced no a subspace (by restricting the metric from the whole space) is the same as the subspace topology arising from the whole space. For full proof, refer: topology from subspace metric equals subspace topology