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
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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).
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Keywords:
<IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\sigma$">-discrete,
network,
symmetric space,
semimetric space
Article copyright:
© Copyright 1970
American Mathematical Society