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Some results on semi-stratifiable spaces. (English) Zbl 1474.54076

Summary: 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 D. Basile et al. [Houston J. Math. 40, No. 1, 255–266 (2014; Zbl 1293.54003)]. As a corollary, we show that \(|X|\le e(X)^{\omega}\) for any semi-stratifiable space \(X\).

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E35 Metric spaces, metrizability

Citations:

Zbl 1293.54003
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References:

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