Some results on semi-stratifiable spaces

Xuan, Wei-Feng, and Song, Yan-Kui. "Some results on semi-stratifiable spaces." Mathematica Bohemica 144.2 (2019): 113-123. <http://eudml.org/doc/294556>.

@article{Xuan2019,
abstract = {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 $DC(\omega _1)$; (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 $\omega $-monolithic star countable extent semi-stratifiable space. If $t(X)=\omega $ and $d(X) \le \omega _1$, 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 $|X| \le e(X)^\{\omega \}$ for any semi-stratifiable space $X$.},
author = {Xuan, Wei-Feng, Song, Yan-Kui},
journal = {Mathematica Bohemica},
keywords = {semi-stratifiable space; separable space; dense subset; feebly compact space; $\omega $-monolithic space; property $DC(\omega _1)$; star countable extent space; cardinal equality; countable chain condition; perfect space; $G^*_\delta $-diagonal},
language = {eng},
number = {2},
pages = {113-123},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some results on semi-stratifiable spaces},
url = {http://eudml.org/doc/294556},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Xuan, Wei-Feng
AU - Song, Yan-Kui
TI - Some results on semi-stratifiable spaces
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 2
SP - 113
EP - 123
AB - 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 $DC(\omega _1)$; (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 $\omega $-monolithic star countable extent semi-stratifiable space. If $t(X)=\omega $ and $d(X) \le \omega _1$, 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 $|X| \le e(X)^{\omega }$ for any semi-stratifiable space $X$.
LA - eng
KW - semi-stratifiable space; separable space; dense subset; feebly compact space; $\omega $-monolithic space; property $DC(\omega _1)$; star countable extent space; cardinal equality; countable chain condition; perfect space; $G^*_\delta $-diagonal
UR - http://eudml.org/doc/294556
ER -