Generalizations of Chainability and Compactness, and the Hypertopologies

TL;DR

This paper introduces generalized concepts of chainability and compactness in metric spaces, examining their properties and relationships with various hypertopologies to deepen understanding of topological structures.

Contribution

It proposes new generalized properties extending chainability and compactness, and analyzes their connections with Hausdorff, Vietoris, and locally finite hypertopologies.

Findings

Generalized properties extend classical notions of chainability and compactness.

Established relations among Hausdorff, Vietoris, and locally finite hypertopologies.

Basic results elucidate the structure and interconnections of these properties.

Abstract

We study two properties for subsets of a metric space. One of them is generalization of chainability, finite chainability, and Menger convexity for metric spaces; while the other is a generalization of compactness. We explore the basic results related to these two properties. Further, in the perspective of these properties, we explore relations among the Hausdorff, Vietoris, and locally finite hypertopologies.