| Title:
|
Remarks on star countable discrete closed spaces (English) |
| Author:
|
Song, Yan-Kui |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
451-460 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming $2^{\aleph _0}=2^{\aleph _1}$, there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable. (English) |
| Keyword:
|
pseudocompact |
| Keyword:
|
normal |
| Keyword:
|
Tychonoff |
| Keyword:
|
star countable |
| Keyword:
|
absolutely star countable |
| Keyword:
|
star countable discrete closed |
| Keyword:
|
absolutely star countable discrete closed space |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 06236423 |
| idMR:
|
MR3073970 |
| DOI:
|
10.1007/s10587-013-0029-x |
| . |
| Date available:
|
2013-07-18T15:00:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143324 |
| . |
| Reference:
|
[1] Aiken, L. P.: Star-covering properties: generalized $\Psi$-spaces, countability conditions, reflection.Topology Appl. 158 (2011), 1732-1737. Zbl 1223.54029, MR 2812483, 10.1016/j.topol.2011.06.032 |
| Reference:
|
[2] Alas, O. T., Junqueira, L. R., Wilson, R. G.: Countability and star covering properties.Topology Appl. 158 (2011), 620-626. Zbl 1226.54023, MR 2765618, 10.1016/j.topol.2010.12.012 |
| Reference:
|
[3] Alas, O. T., Junqueira, L. R., Mill, J. van, Tkachuk, V. V., Wilson, R. G.: On the extent of star countable spaces.Cent. Eur. J. Math. 9 (2011), 603-615. MR 2784032, 10.2478/s11533-011-0018-y |
| Reference:
|
[4] Bonanzinga, M.: Star-Lindelöf and absolutely star-Lindelöf spaces.Questions Answers Gen. Topology. 16 (1998), 79-104. Zbl 0931.54019, MR 1642032 |
| Reference:
|
[5] Engelking, R.: General Topology. Rev. and compl. ed.Sigma Series in Pure Mathematics 6, Heldermann, Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[6] Matveev, M. V.: Absolutely countably compact spaces.Topology Appl. 58 (1994), 81-92. Zbl 0801.54021, MR 1280711, 10.1016/0166-8641(94)90074-4 |
| Reference:
|
[7] Matveev, M. V.: A survey on star-covering properties.Topological Atlas preprint No 330 (1998). |
| Reference:
|
[8] Matveev, M. V.: On space in countable web.arxiv:math/0006199v1 (2000). |
| Reference:
|
[9] Song, Y.-K.: Discretely star-Lindelöf spaces.Tsukuba J. Math. 25 (2001), 371-382. Zbl 1011.54020, MR 1869769, 10.21099/tkbjm/1496164294 |
| Reference:
|
[10] Song, Y.-K.: Closed subsets of absolutely star-Lindelöf spaces II.Commentat. Math. Univ. Carol. 44 (2003), 329-334. Zbl 1100.54017, MR 2026167 |
| Reference:
|
[11] Song, Y.-K.: Remarks on discretely absolutely star-Lindelöf spaces.Czech. Math. J. 58 (2008), 823-831. MR 2455940, 10.1007/s10587-008-0053-4 |
| Reference:
|
[12] Song, Y.-K.: Remarks on countability and star covering properties.Topology Appl. 158 (2011), 1121-1123. Zbl 1220.54008, MR 2794466, 10.1016/j.topol.2011.03.006 |
| Reference:
|
[13] Tall, F. D.: Normality versus collectionwise normality.Handbook of Set-Theoretic Topology K. Kunen and J. E. Vaughan North-Holland, Amsterdam (1984), 685-732. Zbl 0552.54011, MR 0776634 |
| Reference:
|
[14] Douwen, E. K. van, Reed, G. M., Roscoe, A. W., Tree, I. J.: Star covering properties.Topology Appl. 39 (1991), 71-103. MR 1103993, 10.1016/0166-8641(91)90077-Y |
| Reference:
|
[15] Mill, J. van, Tkachuk, V. V., Wilson, R. G.: Classes defined by stars and neighbourhood assignments.Topology Appl. 154 (2007), 2127-2134. MR 2324924, 10.1016/j.topol.2006.03.029 |
| Reference:
|
[16] Vaughan, J. E.: On the product of a compact space with an absolutely countably compact space.Ann. N. Y. Acad. Sci. 788 (1996), 203-208. Zbl 0917.54023, MR 1460834, 10.1111/j.1749-6632.1996.tb36814.x |
| Reference:
|
[17] Yasui, Y., Gao, Z.-M.: Spaces in countable web.Houston J. Math. 25 (1999), 327-335. Zbl 0974.54011, MR 1697629 |
| . |
