  There are several ideal boundaries and completions in General Relativity
sharing the topological property of being sequential, i.e., determined by the
convergence of its sequences and, so, by some limit operator $L$. As emphasized
in a classical article by Geroch, Liang and Wald, some of them have the
property, commonly regarded as a drawback, that there are points of the
spacetime $M$ non $T_1$-separated from points of the boundary $\partial M$.
  Here we show that this problem can be solved from a general topological
viewpoint. In particular, there is a canonical minimum refinement of the
topology in the completion $\overline{M}$ which $T_2$-separates the spacetime
$M$ and its boundary $\partial M$ ---no matter the type of completion one
chooses. Moreover, we analyze the case of sequential spaces and show how the
refined $T_2$-separating topology can be constructed from a modification $L^*$
of the original limit operator $L$. Finally, we particularize this procedure to
the case of the causal boundary and show how the separability of $M$ and
$\partial M$ can be introduced as an abstract axiom in its definition.
