Let $\delta \ge 0$ A metric space $(X,d)$ is $\delta$ hyperbolic if, for any figure $ABC$ in $X$ that is a geodesic triangle with respect to $d$ and for every $P \in \overline{AB}$ there exists a point $Q \in \overline{AC} \cup \overline{BC}$ such that $d(P,Q) \le \delta$

A hyperbolic metric space is a metric space that is $\delta$ hyperbolic for some $\delta \ge 0$

Although a metric space is hyperbolic if it is $\delta$ hyperbolic for some $\delta \ge 0$ one usually tries to find the smallest value of $\delta$ for which a hyperbolic metric space $(X,d)$ is $\delta$ hyperbolic.

A simple example of a hyperbolic metric space is the real line under the usual metric. Given any three points $A,B,C \in \mathbb{R}$ we always have that $\overline{AB} \subseteq \overline{AC} \cup \overline{BC}$ Thus, for any $P \in \overline{AB}$ we can take $Q=P$ Therefore, the real line is 0 hyperbolic. Similar reasoning can be used to show that every real tree is 0 hyperbolic.