  Recently, J. D. Lawson encouraged the domain theory community to consider the
scientific program of developing domain theory in the wider context of $T_0$
spaces instead of restricting to posets. In this paper, we respond to this
calling with an attempt to formulate a topological version of the Scott
Convergence Theorem, i.e., an order-theoretic characterisation of those posets
for which the Scott-convergence $\mathcal{S}$ is topological. To do this, we
make use of the $\mathcal{ID}$ replacement principle to create topological
analogues of well-known domain-theoretic concepts, e.g.,
$\mathcal{I}$-continuous spaces correspond to continuous posets, as
$\mathcal{I}$-convergence corresponds to $\mathcal{S}$-convergence. In this
paper, we consider two novel topological concepts, namely, the
$\mathcal{I}$-stable spaces and the $\mathcal{DI}$ spaces, and as a result we
obtain some necessary (respectively, sufficient) conditions under which the
convergence structure $\mathcal{I}$ is topological.
