# Geodesic in a metric space

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Let $M$ be a metric space with metric $d$. Then, a curve $\gamma:[0,1] \to \R$ (or $(0,1) \to \R$ or $[0,1) \to \R$) 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 $b,c \in U$.