  Given a partial action of a topological group $G$ on a space $X$, we
determine properties $\mathcal P$ which can be extended from $X$ to its
globalization. We treat the cases when $\mathcal P$ is any of the following:
Hausdorff, regular, metrizable, second countable and having invariant metric.
Further, for a normal subgroup $H$ we introduce and study a partial action of
$G/H$ on the orbit space $X/\!\sim,$ applications to invariant metrics and
inverse limits are presented.
