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 | 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.

**1.**A. V. Arkhangel′skiĭ,*Topological function spaces*, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR**1144519****2.**A. V. Arhangel′skii,*Relative topological properties and relative topological spaces*, Proceedings of the International Conference on Convergence Theory (Dijon, 1994), 1996, pp. 87–99. MR**1397067**, 10.1016/0166-8641(95)00086-0**3.**Miroslav Hušek and Jan van Mill (eds.),*Recent progress in general topology*, North-Holland Publishing Co., Amsterdam, 1992. Papers from the Symposium on Topology (Toposym) held in Prague, August 19–23, 1991. MR**1229121****4.**A. V. Arhangel′skii and V. V. Uspenskiĭ,*On the cardinality of Lindelöf subspaces of function spaces*, Comment. Math. Univ. Carolin.**27**(1986), no. 4, 673–676. MR**874660****5.**D. P. Baturov,*Subspaces of function spaces*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**4**(1987), 66–69 (Russian). MR**913076****6.**D. P. Baturov,*Normality in dense subspaces of products*, Topology Appl.**36**(1990), no. 2, 111–116. Seminar on General Topology and Topological Algebra (Moscow, 1988/1989). MR**1068164**, 10.1016/0166-8641(90)90003-K**7.**Ryszard Engelking,*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321****8.**Winfried Just and Jamal Tartir,*A 𝜅-normal, not densely normal Tychonoff space*, Proc. Amer. Math. Soc.**127**(1999), no. 3, 901–905. MR**1469416**, 10.1090/S0002-9939-99-04587-6**9.**D. B. Shakhmatov,*A pseudocompact Tychonoff space all countable subsets of which are closed and 𝐶*-embedded*, Topology Appl.**22**(1986), no. 2, 139–144. MR**836321**, 10.1016/0166-8641(86)90004-0**10.**E.V. Scepin,*Real-valued functions and spaces close to normal*, Sib. Matem. Journ. 13:5 (1972), 1182-1196.

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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