AI 中文总结
研究\(T_0\)空间中拟下极限收敛，引入弱局部超紧空间，证明局部超紧\(T_0\)空间是\(WLH\)空间及相关等价条件，利用拟下极限收敛刻画\(C\)空间和连续偏序集。
AI 中文摘要
本文作者的主要目标是将域论中与下极限收敛和\(\mathcal{QS}\)收敛相关的一些重要结果扩展到\(T_0\)空间的情形。为此，研究了\(T_0\)空间中的拟下极限收敛，并引入了一种新的\(T_0\)空间——弱局部超紧空间（简称\(WLH\)空间）。证明了每个局部超紧\(T_0\)空间是\(WLH\)空间，且\(T_0\)空间\((X,\tau)\)是\(WLH\)空间当且仅当\((X,\tau)\)中的拟下极限收敛是拓扑收敛。还表明\(T_0\)空间\((X,\tau)\)局部超紧当且仅当\((X,\tau)\)中的\(\mathcal{QS}\)收敛与拓扑\(\tau\)中的收敛一致。利用拟下极限收敛给出了\(C\)空间和连续偏序集的几种刻画。
英文摘要
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 $(X, τ)$ is a \emph{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 $T_0$-space $(X,τ)$ is locally hypercompact iff the $\mathcal{QS}$-convergence in $(X,τ)$ coincides with the convergence in the topology $τ$. 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 $σ(P)$. Using the quasi-liminf convergence, we give several characterizations of $C$-spaces and continuous posets.
Comments24 pages, 3 figures
