Product Properties for Generalized Pairwise Lindelöf Spaces
Abstract
In topological spaces, although compactness is satisfying the product invariant properties, but for the Lindelöffness, it is not preserved by the product unless one or more factors are assumed to satisfy additional conditions. Similar results yield for the bitopological spaces, that is, the property of pairwise Lindelöf bitopological spaces is not preserved under the product unless one or more factors are assumed to be satisfy additional conditions, for instance, <img src=image/13492372_01.gif>-spaces. The Cartesian product for arbitrarily many bitopological spaces was defined by Datta in 1972. Since then, many researchers have begun their study for the product bitopological spaces for their reason and direction. In this paper, we shall study finite product of pairwise nearly Lindelöf, pairwise almost Lindelöf and pairwise weakly Lindelöf spaces. We show that, all these generalized pairwise Lindelöf spaces are not preserved under a product by some counterexamples provided. Furthermore, we give some necessary conditions for these three bitopological spaces to be preserved under a finite product. Such condition is that one or more of the spaces has to be <img src=image/13492372_01.gif>-space or the product have to be pairwise weak <img src=image/13492372_01.gif>-space. Another interesting result is that the projection of these generalized pairwise Lindelöf spaces product with <img src=image/13492372_01.gif>-space is a closed map.