Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces

✍ Scribed by Kyriakos Keremedis

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 167 KB
• Volume: 50
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200310084

✦ Synopsis

Abstract

We study within the framework of Zermelo‐Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)