# On $\alpha $-normal and $\beta $-normal spaces

Aleksander V. Arhangel'skii; Lewis D. Ludwig

Commentationes Mathematicae Universitatis Carolinae (2001)

- Volume: 42, Issue: 3, page 507-519
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topArhangel'skii, Aleksander V., and Ludwig, Lewis D.. "On $\alpha $-normal and $\beta $-normal spaces." Commentationes Mathematicae Universitatis Carolinae 42.3 (2001): 507-519. <http://eudml.org/doc/248801>.

@article{Arhangelskii2001,

abstract = {We define two natural normality type properties, $\alpha $-normality and $\beta $-normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every $\beta $-normal, pseudocompact space is countably compact, and show that if $X$ is a dense subspace of a product of metrizable spaces, then $X$ is normal if and only if $X$ is $\beta $-normal. All hereditarily separable spaces are $\alpha $-normal. A space is normal if and only if it is $\kappa $-normal and $\beta $-normal. Central results of the paper are contained in Sections 3 and 4. Several examples are given, including an example (identified by R.Z. Buzyakova) of an $\alpha $-normal, $\kappa $-normal, and not $\beta $-normal space, which is, in fact, a pseudocompact topological group. We observe that under CH there exists a locally compact Hausdorff hereditarily $\alpha $-normal non-normal space (Theorem 3.3). This example is related to the main result of Section 4, which is a version of the famous Katětov’s theorem on metrizability of a compactum the third power of which is hereditarily normal (Corollary 4.3). We also present a Tychonoff space $X$ such that no dense subspace of $X$ is $\alpha $-normal (Section 3).},

author = {Arhangel'skii, Aleksander V., Ludwig, Lewis D.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {normal; $\alpha $-normal; $\beta $-normal; $\kappa $-normal; weakly normal; extremally disconnected; $C_p(X)$; Lindelöf; compact; pseudocompact; countably compact; hereditarily separable; hereditarily $\alpha $-normal; property $wD$; weakly perfect; first countable; normality},

language = {eng},

number = {3},

pages = {507-519},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {On $\alpha $-normal and $\beta $-normal spaces},

url = {http://eudml.org/doc/248801},

volume = {42},

year = {2001},

}

TY - JOUR

AU - Arhangel'skii, Aleksander V.

AU - Ludwig, Lewis D.

TI - On $\alpha $-normal and $\beta $-normal spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2001

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 42

IS - 3

SP - 507

EP - 519

AB - We define two natural normality type properties, $\alpha $-normality and $\beta $-normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every $\beta $-normal, pseudocompact space is countably compact, and show that if $X$ is a dense subspace of a product of metrizable spaces, then $X$ is normal if and only if $X$ is $\beta $-normal. All hereditarily separable spaces are $\alpha $-normal. A space is normal if and only if it is $\kappa $-normal and $\beta $-normal. Central results of the paper are contained in Sections 3 and 4. Several examples are given, including an example (identified by R.Z. Buzyakova) of an $\alpha $-normal, $\kappa $-normal, and not $\beta $-normal space, which is, in fact, a pseudocompact topological group. We observe that under CH there exists a locally compact Hausdorff hereditarily $\alpha $-normal non-normal space (Theorem 3.3). This example is related to the main result of Section 4, which is a version of the famous Katětov’s theorem on metrizability of a compactum the third power of which is hereditarily normal (Corollary 4.3). We also present a Tychonoff space $X$ such that no dense subspace of $X$ is $\alpha $-normal (Section 3).

LA - eng

KW - normal; $\alpha $-normal; $\beta $-normal; $\kappa $-normal; weakly normal; extremally disconnected; $C_p(X)$; Lindelöf; compact; pseudocompact; countably compact; hereditarily separable; hereditarily $\alpha $-normal; property $wD$; weakly perfect; first countable; normality

UR - http://eudml.org/doc/248801

ER -

## References

top- Arhangel'skii A.V., Divisibility and cleavability of spaces, in: W. Gähler, H. Herrlich, G. Preuss, ed-s, Recent Developments in General Topology and its Applications, pp.13-26. Mathematical Research 67, Akademie Verlag, 1992. MR1219762
- Arhangel'skii A.V., Some recent advances and open problems in general topology, Uspekhi Mat. Nauk. 52:5 (1997), 45-70; English translation in Russian Math. Surveys 52:5 (1997), 929-953. (1997) MR1490025
- Arhangel'skii A.V., Topological Function Spaces, Dordrecht; Boston: Kluwer Academic Publishers, 1992. MR1485266
- Arhangel'skii A.V., Normality and dense subspaces, to appear in Proc. Amer. Math. Soc. 2001. Zbl1008.54013MR1855647
- Arhangel'skii A.V., Kočinac L., On a dense ${G}_{\delta}$-diagonal, Publ. Inst. Math. (Beograd) (N.S.) 47 (61) (1990), 121-126. (1990) MR1103538
- Baturov D.P., HASH(0x9f24c00), Subspaces of function spaces, Vestnik Moskov. Univ. Ser. Mat. Mech. 4 (1987), 66-69. (1987) MR0913076
- Baturov D.P., Normality in dense subspaces of products, Topology Appl. 36 (1990), 111-116. (1990) Zbl0695.54018MR1068164
- Blair R.L., Spaces in which special sets are z-embedded, Canad. J. Math. 28:4 (1976), 673-690. (1976) Zbl0359.54009MR0420542
- Bockstein M.F., Un theoréme de séparabilité pour les produits topologiques, Fund. Math. 35 (1948), 242-246. (1948) Zbl0032.19103MR0027503
- Engelking R., General Topology, Heldermann-Verlag, Berlin, 1989. Zbl0684.54001MR1039321
- Heath R.W., On a question of Ljubiša Kočinac, Publ. Inst. Math. (Beograd) (N.S.) 46 (60) (1989), 193-195. (1989) Zbl0694.54021MR1060074
- Jones F.B., Concerning normal and completely normal spaces, Bull. Amer. Math. Soc. 43 (1937), 671-677. (1937) Zbl0017.42902MR1563615
- Jones F.B., Hereditarily separable, non-completely regular spaces, in: Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973), pp.149-152. Lecture Notes in Math. 375, Springer, Berlin, 1974. Zbl0286.54008MR0413044
- Katětov M., Complete normality of Cartesian products, Fund. Math. 35 (1948), 271-274. (1948) MR0027501
- Kočinac L., An example of a new class of spaces, Mat. Vesnik 35:2 (1983), 145-150. (1983) MR0741592
- Mycielski J., $\alpha $-incompactness of ${N}^{\alpha}$, Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. 12 (1964), 437-438. (1964) MR0211871
- Negrepontis S., Banach spaces and topology, in: The Handbook of Set Theoretic Topology, North Holland, 1984, pp.1045-1142. Zbl0832.46005MR0776642
- Nyikos P., Axioms, theorems, and problems related to the Jones lemma, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980), pp.441-449, Academic Press, New York-London, 1981. Zbl0461.54006MR0619071
- Ščepin E.V., Real functions and spaces that are nearly normal, Siberian Math. J. 13 (1972), 820-829. (1972) MR0326656
- Ščepin E.V., On topological products, groups, and a new class of spaces more general than metric spaces, Soviet Math. Dokl. 17:1 (1976), 152-155. (1976) MR0405350
- Singal M.K., Shashi Prabha Arya, Almost normal and almost completely regular spaces, Glasnik Mat. Ser. III 5 (25) (1970), 141-152. (1970) MR0275354

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.