Normality and dense subspaces
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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.References
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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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1855647