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 \(X\).


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


Zbl 1293.54003
Full Text: DOI


[1] Alas, O. T.; Junqueira, L. R.; Mill, J. van; Tkachuk, V. V.; Wilson, R. G., On the extent of star countable spaces, Cent. Eur. J. Math. 9 (2011), 603-615
[2] Alas, O. T.; Junqueira, L. R.; Wilson, R. G., Countability and star covering properties, Topology Appl. 158 (2011), 620-626
[3] Arhangel’skii, A. A.; Buzyakova, R. Z., The rank of the diagonal and submetrizability, Commentat. Math. Univ. Carol. 47 (2006), 585-597
[4] Basile, D.; Bella, A.; Ridderbos, G. J., Weak extent, submetrizability and diagonal degrees, Houston J. Math. 40 (2014), 255-266
[5] Creede, G. D., Concerning semi-stratifiable spaces, Pac. J. Math. 32 (1970), 47-54
[6] Engelking, R., General Topology, Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989)
[7] Gotchev, I. S., Cardinalities of weakly Lindelöf spaces with regular \(G_\kappa\)-diagonals, Avaible at https://scirate.com/arxiv/1504.01785
[8] Gruenhage, G., Generalized metric spaces, Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 423-501 K. Kunen et al
[9] Hodel, R., Cardinal functions. I, Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 1-61 K. Kunen et al
[10] Ikenaga, S., Topological concept between Lindelöf and Pseudo-Lindelöf, Research Reports of Nara National College of Technology 26 (1990), 103-108 Japanese
[11] Juhász, I., Cardinal Functions in Topology, Mathematical Centre Tracts 34. Mathematisch Centrum, Amsterdam (1971)
[12] Rojas-Sánchez, A. D.; Tamariz-Mascarúa, Á., Spaces with star countable extent, Commentat. Math. Univ. Carol. 57 (2016), 381-395
[13] Šapirovskij, B. E., On separability and metrizability of spaces with Souslin’s condition, Sov. Math. Dokl. 13 (1972), 1633-1638 translation from Dokl. Akad. Nauk SSSR 207 1972 800-803\kern0pt
[14] Douwen, E. K. van; Reed, G. M.; Roscoe, A. W.; Tree, I. J., Star covering properties, Topology Appl. 39 (1991), 71-103
[15] Wiscamb, M. R., The discrete countable chain condition, Proc. Am. Math. Soc. 23 (1969), 608-612
[16] Xuan, W. F., Symmetric \(g\)-functions and cardinal inequalities, Topology Appl. 221 (2017), 51-58
[17] Yu, Z., A note on the extent of two subclasses of star countable spaces, Cent. Eur. J. Math. 10 (2012), 1067-1070
[18] Zenor, P., On spaces with regular \(G_\delta \)-diagonal, Pac. J. Math. 40 (1972), 759-763
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