Geodesic in a metric space

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Given data

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.

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