Some results on semi-stratifiable spaces
We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $DC(ømega_1)$; (2) If $X$ is a star countable extent semi-stratifiable...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2019-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/144/2/mb144_2_1.pdf |
| _version_ | 1811299186816057344 |
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| author | Wei-Feng Xuan Yan-Kui Song |
| author_facet | Wei-Feng Xuan Yan-Kui Song |
| author_sort | Wei-Feng Xuan |
| collection | DOAJ |
| description | We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $DC(ømega_1)$; (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; (3) Let $X$ be a $ømega$-monolithic star countable extent semi-stratifiable space. If $t(X)=ømega$ and $d(X) \leømega_1$, then $X$ is hereditarily separable. 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 $|X| \le e(X)^{ømega}$ for any semi-stratifiable space $X$. |
| first_indexed | 2024-04-13T06:31:30Z |
| format | Article |
| id | doaj.art-f2cc79951685420b9fc2eb45d36a6b5b |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-04-13T06:31:30Z |
| publishDate | 2019-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-f2cc79951685420b9fc2eb45d36a6b5b2022-12-22T02:58:08ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362019-07-01144211312310.21136/MB.2018.0043-17MB.2018.0043-17Some results on semi-stratifiable spacesWei-Feng XuanYan-Kui SongWe study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $DC(ømega_1)$; (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; (3) Let $X$ be a $ømega$-monolithic star countable extent semi-stratifiable space. If $t(X)=ømega$ and $d(X) \leømega_1$, then $X$ is hereditarily separable. 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 $|X| \le e(X)^{ømega}$ for any semi-stratifiable space $X$.http://mb.math.cas.cz/full/144/2/mb144_2_1.pdf semi-stratifiable space separable space dense subset feebly compact space $ømega$-monolithic space property $DC(ømega_1)$ star countable extent space cardinal equality countable chain condition perfect space $G^*_\delta$-diagonal |
| spellingShingle | Wei-Feng Xuan Yan-Kui Song Some results on semi-stratifiable spaces Mathematica Bohemica semi-stratifiable space separable space dense subset feebly compact space $ømega$-monolithic space property $DC(ømega_1)$ star countable extent space cardinal equality countable chain condition perfect space $G^*_\delta$-diagonal |
| title | Some results on semi-stratifiable spaces |
| title_full | Some results on semi-stratifiable spaces |
| title_fullStr | Some results on semi-stratifiable spaces |
| title_full_unstemmed | Some results on semi-stratifiable spaces |
| title_short | Some results on semi-stratifiable spaces |
| title_sort | some results on semi stratifiable spaces |
| topic | semi-stratifiable space separable space dense subset feebly compact space $ømega$-monolithic space property $DC(ømega_1)$ star countable extent space cardinal equality countable chain condition perfect space $G^*_\delta$-diagonal |
| url | http://mb.math.cas.cz/full/144/2/mb144_2_1.pdf |
| work_keys_str_mv | AT weifengxuan someresultsonsemistratifiablespaces AT yankuisong someresultsonsemistratifiablespaces |