Remarks on new star-selection principles in topology.

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 U f i n ⁎ (O , O) but does not have the property U f i n ⁎ (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 U 1 ⁎ (O , O) but their products do not have the property U 1 ⁎ (O , O). A Tychonoff example of a space having the property U 1 ⁎ (O , O) which is not star countable is also given. If a space X has the property U 1 ⁎ (O , O) (respectively, U f i n ⁎ (O , O)) and e (X) < ω 1 , then the Alexandroff duplicate A (X) has the property U 1 ⁎ (O , O) (respectively, U f i n ⁎ (O , O)). Finally, we prove that the property U 1 ⁎ (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]. [ABSTRACT FROM AUTHOR]