Lebesgue Number and Total Boundedness

TL;DR

This paper generalizes the Lebesgue number lemma and explores conditions under which chainable metric spaces are totally bounded, introducing a new property that links connectedness, Menger convexity, and compactness.

Contribution

It introduces a generalized Lebesgue number lemma and a new property of metric spaces that unifies connectedness, Menger convexity, and compactness.

Findings

Countably infinite locally finite open covers have Lebesgue numbers in chainable spaces.

A new property generalizes connectedness and Menger convexity.

A metric space with this property equates Atsujiness and compactness.

Abstract

A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric spaces which is a generalization of connectedness and Menger convexity is introduced. It is observed that Atsujiness and compactness are equivalent for a metric space with this introduced property as well as for a chainable metric space.