On $\alpha$-normal and $\beta$-normal spaces
Arhangel'skii, Aleksander V. ; Ludwig, Lewis D.
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 507-519 / Harvested from Czech Digital Mathematics Library
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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).

Publié le : 2001-01-01
Classification:  54D15,  54D65,  54G20 
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     author = {Aleksander V. Arhangel'skii and Lewis D. Ludwig},
     title = {On $\alpha$-normal and $\beta$-normal spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {507-519},
     zbl = {1053.54030},
     mrnumber = {1860239},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119265}
}

Arhangel'skii, Aleksander V.; Ludwig, Lewis D. On $\alpha$-normal and $\beta$-normal spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 507-519. http://gdmltest.u-ga.fr/item/119265/

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