Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A regular Lindelöf semimetric space which has no countable network


Author: E. S. Berney
Journal: Proc. Amer. Math. Soc. 26 (1970), 361-364
MSC: Primary 54.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0270336-7
MathSciNet review: 0270336
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A completely regular semimetric space $ M$ is constructed which has no $ \sigma $-discrete network. The space $ M$ constructed has the property that every subset of $ M$ of cardinality $ {2^{{\aleph _0}}}$ contains a limit point of itself; thus, assuming $ {2^{{\aleph _0}}} = {\aleph _1},M$ is Lindelöf. It is also shown from the same space $ M$ that, assuming $ {2^{{\aleph _0}}} = {\aleph _1}$, there exists a regular Lindelöf semimetric space $ X$ such that $ X \times X$ is not normal (hence not Lindelöf).


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54.40

Retrieve articles in all journals with MSC: 54.40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0270336-7
Keywords: $ \sigma $-discrete, network, symmetric space, semimetric space
Article copyright: © Copyright 1970 American Mathematical Society