Pairwise Semiregular Properties on Generalized Pairwise Lindelof Spaces

Abstract

Let $\left( X,τ_{1},τ_{2}\right) $ be a bitopological space and $% \left( X,τ_{\left( 1,2\right) }^{s},τ_{\left( 2,1\right) }^{s}\right) $ its pairwise semiregularization. Then a bitopological property $\mathcal{P}$\ is called pairwise semiregular provided that $\left( X,τ_{1},τ_{2}\right) $\ has the property $\mathcal{P}$\ if and only if $\left( X,τ_{\left( 1,2\right) }^{s},τ_{\left( 2,1\right) }^{s}\right) $\ has the same property. In this work we study pairwise semiregular property of $\left( i,j\right) $-nearly Lindelöf, pairwise nearly Lindelöf, $\left( i,j\right) $-almost Lindelöf, pairwise almost Lindelöf, $\left( i,j\right) $-weakly Lindelöf and pairwise weakly Lindelöf spaces. We prove that $\left( i,j\right) $-almost Lindel% öf, pairwise almost Lindelöf, $\left( i,j\right) $-weakly Lindelö% f and pairwise weakly Lindelöf are pairwise semiregular properties, on the contrary of each type of pairwise Lindelöf space which are not pairwise semiregular properties.