Further results on new star-selection principles,Topology and its Applications

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Further results on new star-selection principles
Topology and its Applications ( IF 0.6 ) Pub Date : 2021-06-30 , DOI: 10.1016/j.topol.2021.107762
Sumit Singh

Bal and Bhowmik [1] introduced two new star selection principles ${}^{⁎}U_{1} (A, B)$ and ${}^{⁎}U_{f i n} (A, B)$ . Song and Xuan [17] gave an example of a Tychonoff space X which has the property ${}^{⁎}U_{1} (O, O)$ having a regular-closed subset that does not have the property ${}^{⁎}U_{1} (O, O)$ , where $O$ denotes the collection of all open covers of a space X. In this paper, we show that there exists a Tychonoff space with the property ${}^{⁎}U_{1} (O, O)$ (resp., ${}^{⁎}U_{f i n} (O, O)$ ) having a regular-closed $G_{δ}$ -subspace which does not have the property ${}^{⁎}U_{1} (O, O)$ (resp., ${}^{⁎}U_{f i n} (O, O)$ ). We investigate the relationship among the properties ${}^{⁎}U_{1} (O, O)$ , ${}^{⁎}U_{f i n} (O, O)$ , DCCC, and CCC. Finally, we give a combinatorial characterization of Isbell-Mrówka ψ-spaces having the properties ${}^{⁎}U_{1} (O, O)$ and ${}^{⁎}U_{f i n} (O, O)$ , respectively.

中文翻译：

新的选星原则的进一步结果

Bal 和 Bhowmik [1] 介绍了两个新的选星原则 ${}^{⁎}你_{1} (一种, 乙)$ 和 ${}^{⁎}你_{F 一世 n} (一种, 乙)$ . Song 和 Xu [17] 给出了一个具有性质的 Tychonoff 空间X的例子 ${}^{⁎}你_{1} (哦, 哦)$ 有一个不具有该属性的正则闭子集 ${}^{⁎}你_{1} (哦, 哦)$ ，在哪里 $哦$ 表示空间X的所有开覆盖的集合。在本文中，我们证明存在具有性质的 Tychonoff 空间 ${}^{⁎}你_{1} (哦, 哦)$ （分别， ${}^{⁎}你_{F 一世 n} (哦, 哦)$ ) 有一个定期关闭 $G_{δ}$ - 没有属性的子空间 ${}^{⁎}你_{1} (哦, 哦)$ （分别， ${}^{⁎}你_{F 一世 n} (哦, 哦)$ ）。我们调查属性之间的关系 ${}^{⁎}你_{1} (哦, 哦)$ , ${}^{⁎}你_{F 一世 n} (哦, 哦)$ 、DCCC 和 CCC。最后，我们给出斯贝尔- Mrówka的组合特征ψ -spaces具有特性 ${}^{⁎}你_{1} (哦, 哦)$ 和 ${}^{⁎}你_{F 一世 n} (哦, 哦)$ ，分别。

更新日期：2021-07-07

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