Metric spaces induce a topology

A metric space is an ordered pair of a set $S$ and a function $d$, the distance function :

\begin{equation} d : S \times S \to \mathbb{R}_{\geq 0} \end{equation}

such that $d$ obeys the following conditions :

  1. If $d(x,y) = 0$, then $x = y$.
  2. The distance is symmetric : $d(x,y) = d(y,x)$
  3. It obeys the triangle inequality : $$d(x,z) \leq d(x,y) + d(y,z)$$

The metric topology of a metric space is the topology induced by the set of all balls. That is, for every $x \in S$ and every $r \in \mathbb{R}_{> 0}$, a ball is

\begin{equation} B_{x,r} = \{ y | d(x,y) < r \} \end{equation}