Remarks on new star-selection principles in topology

Abstract

In this paper, we study some new selection principles using the star operator which was introduced by Bal and Bhowmik (2017) [1]. We first prove that there exists a space X which has the property Ufin⁎(O,O) but does not have the property Ufin⁎(O,O), where O denotes the collection of all open covers of a space X. We also obtain several examples of spaces having the property U1⁎(O,O) but their products do not have the property U1⁎(O,O). A Tychonoff example of a space having the property U1⁎(O,O) which is not star countable is also given. If a space X has the property U1⁎(O,O) (respectively, Ufin⁎(O,O)) and e(X)<ω1, then the Alexandroff duplicate A(X) has the property U1⁎(O,O) (respectively, Ufin⁎(O,O)). Finally, we prove that the property U1⁎(O,O) is not hereditary with respect to regular closed subsets and every regular paraLindelöf 1-star-Lindelöf space is Lindelöf. The above-mentioned results answer two published open questions from [1].