A $T_2$-separable $g$ boundary and its relation to the $a$ boundary.

We construct a $T_2$-separable $g$ boundary, which we denote by $\tilde{g}$, and establish an explicit embedding of $\tilde{g}$ into the $a$ boundary. We emphasize that rather than seen as an entirely new construction, we view $\tilde{g}$ as a modification of the $g$ boundary, as $\tilde{g}$ is also tied to the central notion of the $g$ boundary, geodesic/curve incompleteness. The $\tilde{g}$ was obtained via a modification of the open sets on the reduced tangent bundle. It is also shown that regular manifolds points remain separated from boundary points upon completion of the spacetime with $\tilde{g}$. We apply our construction to the example of Geroch, Canbin and Wald, and it is seen that manifold points do in fact remain separated from the boundary point. This provides a counterexample to their conjecture that all boundary constructions of a certain broad class will necessarily be topologically pathological.