论文标题
拓扑空间中的渐近紧凑性
Asymptotic compactness in topological spaces
论文作者
论文摘要
欧米茄限制集合在拓扑时间或离散时间内为拓扑半动态系统构建全球吸引子起着基本作用。因此，重要的是要知道何时欧米茄限制集成为非空的紧凑型集合。本文的目的是了解在渐近意义上给定的拓扑空间子集的净网紧凑的机制。为此，我们介绍了子集净渐近紧凑性的概念，并研究与极限集的紧凑性的联系。在本文中，对于给定的非发空子集的网络，我们证明渐近紧凑性和限制集是一个非空的紧凑型集合，从上方收敛的净收敛在可均匀的空间中等效。我们还通过引入定向集的顺序概念来研究渐近紧凑性的概念的顺序版本。
The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we prove that the asymptotic compactness and the property that the limit set is a nonempty compact set to which the net converges from above are equivalent in uniformizable spaces. We also study the sequential version of the notion of asymptotic compactness by introducing the notion of sequentiality of directed sets.
