| Title:
|
The rank of the diagonal and submetrizability (English) |
| Author:
|
Arhangel'skii, A. V. |
| Author:
|
Buzyakova, R. Z. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
585-597 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Several topological properties lying between the submetrizability and the $G_\delta $-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_\delta $-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_\delta $-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank $5$-diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a $G_\delta $-diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular $G_\delta $-diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed. (English) |
| Keyword:
|
$G_\delta $-diagonal |
| Keyword:
|
rank $k$-diagonal |
| Keyword:
|
submetrizability |
| Keyword:
|
condensation |
| Keyword:
|
regular $G_\delta $-diagonal |
| Keyword:
|
zero-set diagonal |
| Keyword:
|
Čech-completeness |
| Keyword:
|
pseudocompact space |
| Keyword:
|
Moore space |
| Keyword:
|
Mrowka space |
| Keyword:
|
bounded subset |
| Keyword:
|
extent |
| Keyword:
|
Souslin number |
| MSC:
|
54D20 |
| MSC:
|
54E99 |
| MSC:
|
54F99 |
| idZBL:
|
Zbl 1150.54335 |
| idMR:
|
MR2337413 |
| . |
| Date available:
|
2009-05-05T16:59:46Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119619 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: External bases of sets lying in bicompacta.Soviet Math. Dokl. 1 (1960), 573-574. MR 0119188 |
| Reference:
|
[2] Arhangel'skii A.V., Burke D.K.: Spaces with a regular $G_\delta $-diagonal.Topology Appl. 153 (2006), 11 1917-1929. Zbl 1117.54004, MR 2227036 |
| Reference:
|
[3] Burke D.K.: Covering properties.in: Handbook of Set-theoretic Topology, K. Kunen and J. Vaughan, eds., North-Holland, Amsterdam, 1984, pp.347-422. Zbl 0569.54022, MR 0776628 |
| Reference:
|
[4] Bella A.: Remarks on the metrizability degree.Boll. Un. Mat. Ital. A (7) 1 3 (1987), 391-396. Zbl 0639.54003, MR 0923425 |
| Reference:
|
[5] Buzyakova R.Z.: Observations on spaces with zero-set or regular $G_\delta $-diagonals.Comment. Math. Univ. Carolin. 46 3 (2005), 169-473. MR 2174525 |
| Reference:
|
[6] Buzyakova R.Z.: Cardinalities of ccc-spaces with regular $G_\delta $-diagonals.Topology Appl. 153 (2006), 11 1696-1698. Zbl 1094.54001, MR 2227022 |
| Reference:
|
[7] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[8] Gittings R.F.: Subclasses of $p$-spaces and strict $p$-spaces.Topology Proc. 3 2 (1978), 335-345. MR 0540500 |
| Reference:
|
[9] Gittings R.F.: Characterizations of spaces by embeddings in $\beta X$.Topology Appl. 11 2 (1980), 149-159. Zbl 0423.54018, MR 0572370 |
| Reference:
|
[10] Hodel R.E.: Moore spaces and $w\Delta $-spaces.Pacific J. Math. 38 3 (1971), 641-652. Zbl 0219.54024, MR 0307169 |
| Reference:
|
[11] Ishii T.: On $wM$-spaces, II.Proc. Japan Acad. 46 (1970), 11-15. Zbl 0198.27203 |
| Reference:
|
[12] Martin H.W.: Contractibility of topological spaces onto metric spaces.Pacific J. Math. 61 1 (1975), 209-217. Zbl 0304.54026, MR 0410685 |
| Reference:
|
[13] McArthur W.G.: $G_\delta $-diagonals and metrization theorems.Pacific J. Math. 44 (1973), 613-617. Zbl 0256.54016, MR 0317288 |
| Reference:
|
[14] Reed G.M.: On normality and countable paracompactness.Fund. Math. 110 (1980), 145-152. Zbl 0446.54031, MR 0600588 |
| Reference:
|
[15] Šapirovskij B.E.: On separability and metrizability of spaces with Souslin condition.Soviet Math. Dokl. 13 6 (1972), 1633-1638. |
| Reference:
|
[16] Zenor P.: On spaces with regular $G_\delta $-diagonals.Pacific J. Math. 40 (1972), 759-763. MR 0307195 |
| . |
