Completeness in semimetric spaces

Fred Galvin and Samuel David Shore

Vol. 113 (1984), No. 1, 67–75

DOI: 10.2140/pjm.1984.113.67

Abstract

We establish interrelationships between Cauchy completeness, McAuley’s notions of strong and weak completeness, and Moore completeness in semimetrizable spaces as well as developable, 1-continuously, or continuously semimetrizable spaces.

Our main result shows that a semimetrizable space may admit one semimetric which is Cauchy complete and a second semimetric which is developable, and yet will not admit a semimetric which is simultaneously Cauchy complete and developable.

Mathematical Subject Classification 2000

Primary: 54E25

Secondary: 54E30

Milestones

Received: 12 April 1982

Revised: 12 May 1983

Published: 1 July 1984