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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
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
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 $X$.
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