Communications of the
Korean Mathematical Society CKMS

Using $(r,s)$-preopen sets [14] and pre-$\omega_t$-closures [6], a new kind of covering property $P_{(\omega_r,s)}^t$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-$(\omega_r,s)t$-$\theta_f$-cluster sets and $(r,s)t$-$\theta_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-$(\omega_r,s)t$-$\theta_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P_{(\omega_r,s)}^t$-closedness has also been established in terms of pre-$(\omega_r,s)t$-$\theta_f$-cluster sets.

Keywords: pre-$\omega$-closure, $\theta$-closure, pre-$(\omega_r,s)$-open, $P_{(\omega_r,s)}^t$-closed, $P_{(r,s)}^t$-closed, $(\omega_r,s)t$-regular, pre-$(\omega_r,s)$-$\theta_t$-complete adherent, pre-$(\omega_r,s)t$-$\theta_f$-cluster set, $(r,s)t$-$\theta_f$-p

MSC numbers: 54A10, 54E55, 54D20, 54A20, 54C99, 54C60