Mathematica Bohemica, Vol. 144, No. 2, pp. 113-123, 2019
Some results on semi-stratifiable spaces
Wei-Feng Xuan, Yan-Kui Song
Received May 9, 2017. Published online April 12, 2018.
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 $X$.
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Affiliations: Wei-Feng Xuan, School of Statistics and Mathematics, Nanjing Audit University, Nanjing, China, 211815, e-mail: wfxuan@nau.edu.cn; Yan-Kui Song, Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, China, 210046, e-mail: songyankui@njnu.edu.cn
