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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Normality and dense subspaces


Author: A. V. Arhangel'skii
Journal: Proc. Amer. Math. Soc. 130 (2002), 283-291
MSC (1991): Primary 54C35, 54D15, 54D20
Published electronically: May 25, 2001
MathSciNet review: 1855647
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Abstract:

In the first section of this paper, using certain powerful results in $C_{p}$-theory, we show that there exists a nice linear topological space $X$of weight $\omega _{1}$ such that no dense subspace of $X$ is normal. In the second and third sections a natural generalization of normality, called dense normality, is considered. In particular, it is shown in section 2 that the space $R^{c}$ is not normal on some countable dense subspace of it, while it is normal on some other dense subspace. An example of a Tychonoff space $X$, which is not densely normal on a dense separable metrizable subspace, is constructed. In section 3, a link between dense normality and relative countable compactness is established. In section 4 the result of section 1 is extended to densely normal spaces.


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Additional Information

A. V. Arhangel'skii
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
Email: arhangel@bing.math.ohiou.edu, arhala@arhala.mccme.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06051-8
PII: S 0002-9939(01)06051-8
Keywords: Normal space, extent, Lindel\"{o}f number, Souslin number, $C_{p}$-theory, densely normal, $\kappa $-normal, $X$ normal on $Y$, $A$ concentrated on $Y$, pseudocompact, relative countable compactness, locally connected
Received by editor(s): July 13, 1998
Received by editor(s) in revised form: June 6, 2000
Published electronically: May 25, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society