## 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

Zbl 1293.54003
Full Text:

### References:

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