| Title:
|
Some results on semi-stratifiable spaces (English) |
| Author:
|
Xuan, Wei-Feng |
| Author:
|
Song, Yan-Kui |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
113-123 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: \endgraf (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $DC(\omega _1)$; \endgraf (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; \endgraf (3) Let $X$ be a $\omega $-monolithic star countable extent semi-stratifiable space. If $t(X)=\omega $ and $d(X) \le \omega _1$, then $X$ is hereditarily separable. \endgraf Finally, we prove that for any $T_1$-space $X$, $|X| \le L(X)^{\Delta (X)}$, which gives a partial answer to a question of Basile, Bella, and Ridderbos (2011). As a corollary, we show that $|X| \le e(X)^{\omega }$ for any semi-stratifiable space $X$. (English) |
| Keyword:
|
semi-stratifiable space |
| Keyword:
|
separable space |
| Keyword:
|
dense subset |
| Keyword:
|
feebly compact space |
| Keyword:
|
$\omega $-monolithic space |
| Keyword:
|
property $DC(\omega _1)$ |
| Keyword:
|
star countable extent space |
| Keyword:
|
cardinal equality |
| Keyword:
|
countable chain condition |
| Keyword:
|
perfect space |
| Keyword:
|
$G^*_\delta $-diagonal |
| MSC:
|
54D20 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 07088839 |
| idMR:
|
MR3974181 |
| DOI:
|
10.21136/MB.2018.0043-17 |
| . |
| Date available:
|
2019-06-21T11:31:17Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147752 |
| . |
| Reference:
|
[1] 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. Zbl 1246.54017, MR 2784032, 10.2478/s11533-011-0018-y |
| 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] Arhangel'skii, A. A., Buzyakova, R. Z.: The rank of the diagonal and submetrizability.Commentat. Math. Univ. Carol. 47 (2006), 585-597. Zbl 1150.54335, MR 2337413 |
| Reference:
|
[4] Basile, D., Bella, A., Ridderbos, G. J.: Weak extent, submetrizability and diagonal degrees.Houston J. Math. 40 (2014), 255-266. Zbl 1293.54003, MR 3210565 |
| Reference:
|
[5] Creede, G. D.: Concerning semi-stratifiable spaces.Pac. J. Math. 32 (1970), 47-54. Zbl 0189.23304, MR 0254799, 10.2140/pjm.1970.32.47 |
| Reference:
|
[6] Engelking, R.: General Topology.Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[7] Gotchev, I. S.: Cardinalities of weakly Lindelöf spaces with regular $G_\kappa$-diagonals.Avaible at https://scirate.com/arxiv/1504.01785. MR 3958260 |
| Reference:
|
[8] Gruenhage, G.: Generalized metric spaces.Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 423-501 K. Kunen et al. Zbl 0555.54015, MR 0776629, 10.1016/B978-0-444-86580-9.50013-6 |
| Reference:
|
[9] Hodel, R.: Cardinal functions. I.Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 1-61 K. Kunen et al. Zbl 0559.54003, MR 0776620 |
| Reference:
|
[10] Ikenaga, S.: Topological concept between Lindelöf and Pseudo-Lindelöf.Research Reports of Nara National College of Technology 26 (1990), 103-108 Japanese. |
| Reference:
|
[11] Juhász, I.: Cardinal Functions in Topology.Mathematical Centre Tracts 34. Mathematisch Centrum, Amsterdam (1971). Zbl 0224.54004, MR 0340021 |
| Reference:
|
[12] Rojas-Sánchez, A. D., Tamariz-Mascarúa, Á.: Spaces with star countable extent.Commentat. Math. Univ. Carol. 57 (2016), 381-395. Zbl 06674888, MR 3554518, 10.14712/1213-7243.2015.176 |
| Reference:
|
[13] Šapirovskij, B. E.: On separability and metrizability of spaces with Souslin's condition.Sov. Math. Dokl. 13 (1972), 1633-1638 translation from Dokl. Akad. Nauk SSSR 207 1972 800-803\kern0pt. Zbl 0268.54007, MR 0322801 |
| Reference:
|
[14] Douwen, E. K. van, Reed, G. M., Roscoe, A. W., Tree, I. J.: Star covering properties.Topology Appl. 39 (1991), 71-103. Zbl 0743.54007, MR 1103993, 10.1016/0166-8641(91)90077-Y |
| Reference:
|
[15] Wiscamb, M. R.: The discrete countable chain condition.Proc. Am. Math. Soc. 23 (1969), 608-612. Zbl 0184.26304, MR 0248744, 10.2307/2036596 |
| Reference:
|
[16] Xuan, W. F.: Symmetric $g$-functions and cardinal inequalities.Topology Appl. 221 (2017), 51-58. Zbl 1376.54026, MR 3624444, 10.1016/j.topol.2017.02.064 |
| Reference:
|
[17] Yu, Z.: A note on the extent of two subclasses of star countable spaces.Cent. Eur. J. Math. 10 (2012), 1067-1070. Zbl 1243.54042, MR 2902235, 10.2478/s11533-012-0030-x |
| Reference:
|
[18] Zenor, P.: On spaces with regular $G_\delta $-diagonal.Pac. J. Math. 40 (1972), 759-763. Zbl 0213.49504, MR 0307195, 10.2140/pjm.1972.40.759 |
| . |
