In this paper we study topological properties of Lyapunov stable sets for semigroup actions on Tychonoff spaces. We show that the stability and the asymptotical stability of a compact set is characterized by the stability and the asymptotical stability of its components, respectively. We also present a characterization of the stability of a compact set by means of ω-limit sets, prolongations, prolongational limit sets and the existence of a fundamental system of invariant neighborhoods of it.

中文翻译：

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半群作用的延长集和稳定集的拓扑性质

在本文中，我们研究Tychonoff空间上半群作用的Lyapunov稳定集的拓扑性质。我们表明，紧集的稳定性和渐近稳定性分别以其组件的稳定性和渐近稳定性为特征。我们还通过ω-极限集，延长，延长极限集以及它的不变邻域的基本系统的存在来描述紧集的稳定性。