Some results on semi-stratifiable spaces

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 $DC(ømega_1)$; (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 $X$.

Keywords

• 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