  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 by proving a topological parallel of a 2005 result due to B. Zhao and
D. Zhao, i.e., an order-theoretic characterisation of those posets for which
the lim-inf convergence is topological. We do this by adopting a recent
approach due to D. Zhao and W. K. Ho by replacing directed subsets with
irreducible sets. As a result, we formulate a new convergence class on $T_0$
spaces called Irr-convergence and established that this convergence class
$\mathcal{I}$ on a $k$-bounded sober space $X$ is topological if and only if
$X$ is Irr-continuous.
