Before revision: An approach to asymptotic finite approximations of geometrical spaces is presented  .
After revision: An approach to asymptotic finite approximations of geometrical spaces is presented . It is based on the well known inverse sequences of finite graphs that determine topological spaces. If a pattern of the construction is sufficiently regular and uniform, then the notions of geodesic and curvature can be defined in the space as the limits of their finite versions in the graphs. This gives rise to consider the graphs as finite approximations of the geometry of the space .