Lebesgue number and total boundedness

Abstract

A generalization of the Lebesgue number lemma is obtained. For a metric space X in the class of strongly metrizable spaces, sufficient conditions for each open cover of X with a Lebesgue number has a finite subcover are 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 for 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.

Keywords

• locally finite open cover
• atsuji space
• chainability