Geodesic in a metric space

Definition

Given data

Let $M$ be a metric space with metric $d$. Then, a curve $\gamma :[0,1]\to \mathbb{R}$ (or $(0,1)\to \mathbb{R}$ or $[0,1)\to \mathbb{R}$) is termed a geodesic if for any $a\in [0,1]$, there exists an open subset $U$ of $a$ in $[0,1]$ such that the restriction of $\gamma$ to $[b,c]$ is the shortest path between them in the metric space, for any $b,c\in U$.

If it is true that the geodesic is the shortest path between its endpoints, it is termed a minimizing geodesic.