Characterization of $T_0$-spaces for quasi-liminf convergence being topological

The authors' primary goal in this paper is to extend some important results related to the liminf-convergence and $\mathcal{QS}$-convergence in domain theory to the setting of $T_0$-spaces. To that end, we study the quasi-liminf convergence in $T_0$-spaces and introduce a new kind of $T_0$-spaces --- weakly locally hypercompact spaces (shortly \emph{WLH}-spaces). It is proved that every locally hypercompact $T_0$-space is a \emph{WLH}-space, and a $T_0$-space $(X, \tau)$ is a \emph{WLH}-space iff the quasi-liminf convergence in $(X, \tau)$ is topological. Hence the quasi-liminf convergence in a locally hypercompact space is topological, and for a quasicontinuous poset $P$, the quasi-liminf convergence is topological and agrees with convergence in the Lawson topology $\lambda(P)$. We also show that a $T_0$-space $(X,\tau)$ is locally hypercompact iff the $\mathcal{QS}$-convergence in $(X,\tau)$ coincides with the convergence in the topology $\tau$. Therefore, a poset $P$ is quasicontinuous iff $\mathcal{QS}$-convergence in the Scott space of $P$ is topological iff $\mathcal{QS}$-convergence coincides with convergence in the Scott topology $\sigma (P)$. Using the quasi-liminf convergence, we give several characterizations of $C$-spaces and continuous posets.