2026/07/28 by Xinpeng Wen, Xiaoquan Xu
#math.GN
The authors' primary goal in this paper is to extend some important results related to the liminf-convergence and QS-convergence in domain theory to the setting of T0-spaces. To that end, we study the quasi-liminf convergence in T0-spaces and introduce a new kind of T0-spaces --- weakly locally hypercompact spaces (shortly WLH-spaces). It is proved that every locally hypercompact T0-space is a WLH-space, and a T0-space (X, τ) is a WLH-space iff the quasi-liminf convergence in (X, τ) 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 λ(P). We also show that a T0-space (X,τ) is locally hypercompact iff the QS-convergence in (X,τ) coincides with the convergence in the topology τ. Therefore, a poset P is quasicontinuous iff QS-convergence in the Scott space of P is topological iff QS-convergence coincides with convergence in the Scott topology σ(P). Using the quasi-liminf convergence, we give several characterizations of C-spaces and continuous posets.