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

@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 -