  The global extendibility of smooth causal geodesically incomplete spacetimes
is investigated. Denote by $\gamma$ one of the incomplete non-extendible causal
geodesics of a causal geodesically incomplete spacetime $(M,g_{ab})$. First, it
is shown that it is always possible to select a synchronised family of causal
geodesics $\Gamma$ and an open neighbourhood $\mathcal{U}$ of a final segment
of $\gamma$ in $M$ such that $\mathcal{U}$ is comprised by members of $\Gamma$,
and suitable local coordinates can be defined everywhere on $\mathcal{U}$
provided that $\gamma$ does not terminate either on a tidal force tensor
singularity or on a topological singularity. It is also shown that if, in
addition, the spacetime, $(M,g_{ab})$, is globally hyperbolic, and the
components of the curvature tensor, and its covariant derivatives up to order
$k-1$ are bounded on $\mathcal{U}$, and also the line integrals of the
components of the $k^{th}$-order covariant derivatives are finite along the
members of $\Gamma$---where all the components are meant to be registered with
respect to a synchronised frame field on $\mathcal{U}$---then there exists a
$C^{k-}$ extension $\Phi: (M,g_{ab}) \rightarrow
(\widehat{M},\widehat{g}_{ab})$ so that for each $\bar\gamma\in\Gamma$, which
is inextendible in $(M,g_{ab})$, the image, $\Phi\circ\bar\gamma$, is
extendible in $(\widehat{M},\widehat{g}_{ab})$. Finally, it is also proved that
whenever $\gamma$ does terminate on a topological singularity $(M,g_{ab})$
cannot be generic.
