| Title:
|
On $\alpha$-normal and $\beta$-normal spaces (English) |
| Author:
|
Arhangel'skii, A. V. |
| Author:
|
Ludwig, L. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
507-519 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
normal |
| Keyword:
|
$\alpha$-normal |
| Keyword:
|
$\beta$-normal |
| Keyword:
|
$\kappa$-normal |
| Keyword:
|
weakly normal |
| Keyword:
|
extremally disconnected |
| Keyword:
|
$C_p(X)$ |
| Keyword:
|
Lindelöf |
| Keyword:
|
compact |
| Keyword:
|
pseudocompact |
| Keyword:
|
countably compact |
| Keyword:
|
hereditarily separable |
| Keyword:
|
hereditarily $\alpha $-normal |
| Keyword:
|
property $wD$ |
| Keyword:
|
weakly perfect |
| Keyword:
|
first countable |
| MSC:
|
54D15 |
| MSC:
|
54D65 |
| MSC:
|
54G20 |
| idZBL:
|
Zbl 1053.54030 |
| idMR:
|
MR1860239 |
| . |
| Date available:
|
2009-01-08T19:12:16Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119265 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
