论文标题
紧凑和对称井订单
Compactness and Symmetric Well Orders
论文作者
论文摘要
我们介绍并研究了Stäckel1907年有限集的拓扑版本，其目的是获得一个有趣的概念，该概念表征了通常的紧凑性（或它的紧密变体）。如果在$ x $上有一些线性订购$ \ prec $，则将$ T_2 $拓扑空间$（x，τ）$定义为stäckel-compact，以便每个非空的$τ$ closed seet都包含$ \ prec-prec $ -least $ - least and $ \ prec-prec-prec-great-greatest element。我们发现紧凑的空间是stäckel-compact，但不相反，而stäckel-compact空间则是次数紧凑的。 Stäckel-compactness与可计的紧凑性的等效性保持开放，但我们的主要结果是，在ZFC下，这种等效性在Cantor-Bendixson Rank $ <ω_2$的散落空间中。在v = l下，等价在所有散射的空间中都保留。
We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X, τ)$ to be Stäckel-compact if there is some linear ordering $\prec$ on $X$ such that every non-empty $τ$-closed set contains a $\prec$-least and a $\prec$-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank $< ω_2$ under ZFC. Under V=L, the equivalence holds in all scattered spaces.
