Asymptotic compactness in topological spaces

TL;DR

This paper investigates the conditions under which nets of subsets in topological spaces are asymptotically compact, linking this property to the compactness of omega limit sets and introducing new notions of asymptotic compactness and sequentiality.

Contribution

It introduces the concept of asymptotic compactness for nets of subsets and establishes its equivalence with limit set compactness in uniformizable spaces, also exploring sequential versions.

Findings

Asymptotic compactness is equivalent to the compactness of limit sets in uniformizable spaces.

Introduces the notion of sequentiality of directed sets for sequential asymptotic compactness.

Provides conditions under which nets of subsets converge to nonempty compact omega limit sets.

Abstract

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.