Metric space

A metric space is pair: a set $X$ together with a real valued function $d: X \times X \longrightarrow {\bf R}$ (called a metric) such that, for every $x,y,z \in X$,

• $d(x,y) \geq 0$, with equality if and only if $x=y$;
• $d(x,y) = d(y,x)$;
• $d(x,z) \leq d(x,y) + d(y,z)$.

The last condition is called the triangle inequality.

For $a \in X$ and $\epsilon > 0$, the open ball around $a$ of radius $\epsilon$ is the set $$B_\epsilon(a) := \{x \in X \mid d(x,a) < \epsilon\}.$$ The topology on $X$ generated by these sets (as a basis of topology) is called the metric topology. This topology is Hausdorff.

The Euclidean space ${\bf R}^n$ is a metric space with the metric defined by $d(x,y) = ||x-y||$. Similarly, all normed spaces are metric spaces.