Normality and dense subspaces

Author:
A. V. Arhangel'skii

Journal:
Proc. Amer. Math. Soc. **130** (2002), 283-291

MSC (1991):
Primary 54C35, 54D15, 54D20

DOI:
https://doi.org/10.1090/S0002-9939-01-06051-8

Published electronically:
May 25, 2001

MathSciNet review:
1855647

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In the first section of this paper, using certain powerful results in -theory, we show that there exists a nice linear topological space of weight such that no dense subspace of 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 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 , 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:
https://doi.org/10.1090/S0002-9939-01-06051-8

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