${P}_{( \omega_{r},s)}^{t}$-closed spaces and pre-$(\omega_{r},{s}){t}$-$\theta_f$-cluster sets
Commun. Korean Math. Soc. 2011 Vol. 26, No. 1, 135-149
https://doi.org/10.4134/CKMS.2011.26.1.135
Printed March 1, 2011
Bin Mostakim Uzzal Afsan and Chanchal Kumar Basu
Sripat Singh College, West Bengal State University
Abstract : 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
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