标题： 集合上的Sobert拓扑 显示英文标题

标题： Sober topologies on a set

摘要： 所有在集合 X 上的拓扑的集合按照包含序构成一个完整的格，这一结构已经被许多研究人员研究过。 斯博性是非豪斯多夫拓扑中的核心且被广泛研究的性质之一。 这一性质在表征交换环的谱空间以及由其开集格确定的拓扑空间中起着关键作用。 在本文中，我们研究给定集合上所有拓扑的完整格中斯博拓扑的状态。 要证明的主要结果包括：(1) 每个 T1 拓扑都是某些斯博拓扑的并；(2) 每个拓扑都是某些斯博拓扑的交；(3) 所有斯博拓扑的集合是定向完全的；(4) 每个亚历山德罗夫-离散拓扑都是某些斯博亚历山德罗夫-离散拓扑的交； (5) 最小的斯博拓扑恰好是上完备链的斯科特拓扑；(6) 将构造一个例子来说明一个递减序列的豪斯多夫拓扑的交不一定是斯博的。 显示英文摘要

摘要： The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff topology. This property plays a crucial role in characterizing the spectral spaces of commutative rings and topological spaces determined by their lattices of open sets. In this paper, we investigate the statute of sober topologies in the complete lattice of all topologies on a given set. The main results to be proved include: (1) every T1 topology is the join of some sober topologies; (2) every topology is the meet of some sober topologies; (3) the set of all sober topologies is directed complete; (4) every Alexanderoff - discrete topology is the meet of some sober Alexanderoff - discrete topologies; (5) the minimal sober topologies are exactly the Scott topologies of sup-complete chains; (6) an example will be constructed to show that the intersection of a decreasing sequence of Hausdorff topologies need not be sober.