论文标题
上越野拓扑的巧合和Scott拓扑
Coincidence of the upper Vietoris topology and the Scott topology
论文作者
论文摘要
对于$ t_0 $ x $ x $，让$ \ mk（x）$是所有紧凑型饱和集的poset，并带有反向包含订单的$ x $。空间$ x $据说具有属性q，如果对于任何$ k_1，k_2 \ in \ mk（x）$，$ k_2 \ ll k_1 $ in $ \ mk（x）$如果{} f $ k_2 \ subseteq \ subseteq \ ii〜 \！k_1 $。在本文中，我们在过滤$ x $的滤清器中提供了多个连接，清醒的$ x $，$ x $的本地紧凑性，$ x $的核心紧凑性，$ x $的属性q的属性q $ x $，上越越野拓扑的巧合，$ \ mk（x）$的苏特拓扑和scott topology and scott topology of $ \ mk（x）$，以及$ x \ x \ x \ \ \ \ \ \ \ \ \ \ f。 σ〜 \！\！ \ mk（x）$（其中$ σ〜 \！\！\ mk（x）$是$ \ mk（x）$）的Scott空间。结果表明，对于过滤良好的空间$ x $，其Smyth Power Space $ p_s（x）$是首先计算的，以下三个属性是等效的：$ x $的本地紧凑性，$ x $的核心紧凑性和$ \ \ mk（x）$的连续性。还证明，对于首先计算的$ t_0 $ x $ x $，其中$ k $的最小元素对于任何紧凑的饱和子集$ k $ of $ x $都是可计数的，smyth power space $ p_s $ p_s（x）$是首先计算的。对于Alexandroff Double Circle $ Y $，这是Hausdorff且首次计数，我们表明其Smyth Power Space $ P_S（y）$不可用。
For a $T_0$ space $X$, let $\mk (X)$ be the poset of all compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if for any $K_1, K_2\in \mk (X)$, $K_2\ll K_1$ in $\mk (X)$ if{}f $K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$, the core compactness of $X$, the property Q of $X$, the coincidence of the upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of $x\mapsto\ua x : X \longrightarrow Σ~\!\! \mk (X)$ (where $Σ~\!\! \mk (X)$ is the Scott space of $\mk (X)$). It is shown that for a well-filtered space $X$ for which its Smyth power space $P_S(X)$ is first-countable, the following three properties are equivalent: the local compactness of $X$, the core compactness of $X$ and the continuity of $\mk (X)$. It is also proved that for a first-countable $T_0$ space $X$ in which the set of minimal elements of $K$ is countable for any compact saturated subset $K$ of $X$, the Smyth power space $P_S(X)$ is first-countable. For the Alexandroff double circle $Y$, which is Hausdorff and first-countable, we show that its Smyth power space $P_S(Y)$ is not first-countable.
