Browse > Article
http://dx.doi.org/10.4134/CKMS.2011.26.1.135

Pt(ωr,s)-CLOSED SPACES AND PRE-(ωr,s)t-θf-CLUSTER SETS  

Afsan, Bin Mostakim Uzzal (DEPARTMENT OF MATHEMATICS SRIPAT SINGH COLLEGE)
Basu, Chanchal Kumar (DEPARTMENT OF MATHEMATICS WEST BENGAL STATE UNIVERSITY)
Publication Information
Communications of the Korean Mathematical Society / v.26, no.1, 2011 , pp. 135-149 More about this Journal
Abstract
Using (r, s)-preopen sets [14] and pre-${\omega}_t$-closures [6], a new kind of covering property $P^t_{({\omega}_r,s)}$-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 $P^t_{({\omega}_r,s)}$-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; $P^t_{({\omega}_r,s)}$-closed; $P^t_{(r,s)}$-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-${\theta}_f$-precluster set; pre-(${\omega}_r$, s)t-${\theta}_f$-irresolute; strongly pre-(${\omega}_r$, s)t-${\theta}_f$-continuous; (r, s)f-preopen; pairwise (r, s)-Urysohn; almost regular;
Citations & Related Records

Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963), 71-89.   DOI
2 N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41.   DOI   ScienceOn
3 S. N. Maheshwari and R. Prasad, Some new separation axioms in bitopological spaces, Mat. Vesnik 12(27) (1975), no. 2, 159-162.
4 S. N. Maheshwari and R. Prasad, Semi-open sets and semicontinuous functions in bitopological spaces, Math. Notae 26 (1977/78), 29-37.
5 A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt No. 53 (1982), 47-53.
6 M. N. Mukherjee and A. Debray, On S-cluster sets and S-closed spaces, Int. J. Math. Math. Sci. 23 (2000), no. 9, 597-603.   DOI   ScienceOn
7 O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970.   DOI
8 T. Noiri, A. Al-Omari, and M. S. M. Noorani, Weak forms of ${\omega}-open$ sets and decompositions of continuity, Eur. J. Pure Appl. Math. 2 (2009), no. 1, 73-84.
9 T. Noiri, A. Al-Omari, and M. S. M. Noorani, Slightly ${\omega}-continuous$ functions, Fasc. Math. No. 41 (2009), 97-106.
10 A. Al-Omari and M. S. M. Noorani, $Contra-{\omega}-continuous$ and almost $contra-{\omega}-continuous$, Int. J. Math. Math. Sci. 2007 (2007), Art. ID 40469, 13 pp.
11 A. Al-Omari and M. S. M. Noorani, Regular generalized ${\omega}-closed$ sets, Int. J. Math. Math. Sci. 2007 (2007), Art. ID 16292, 11 pp.
12 M. K. Singal and S. Prabha Arya, On almost-regular spaces, Glasnik Mat. Ser. III 4 (24) (1969), 89-99.
13 C. K. Basu and M. K. Cohosh, ${\beta}-closed$ spaces and ${\beta}-{\theta}-subclosed$ graphs, Eur. J. Pure Appl. Math. 1 (2008), no. 3, 40-50.
14 C. K. Basu and M. K. Cohosh, Locally ${\beta}-closed$ spaces, Eur. J. Pure Appl. Math. 2 (2009), no. 1, 85-96.
15 C. K. Basu and B. M. Uzzal Afsan, $P_{\omega}-Closedness$ and Its generalization with respect to a grill, communicated.
16 C. K. Basu, B. M. Uzzal Afsan, and S. S. Mandal, Degeneracy of some cluster sets, Italian J. of Pure and Appl. Math. (accepted and to appear).
17 H. Z. Hdeib, ${\omega}-closed$ mappings, Rev. Colombiana Mat. 16 (1982), no. 1-2, 65-78.
18 C. K. Basu, B. M. Uzzal Afsan, and S. S. Mandal, Functions with strongly ${\beta}-{\theta}-closed$ graphs, Mathematica (Cluj), Tome 51(74), No. 2 (Dec 2009), 119-128.
19 S. Bose and D. Sinha, Almost open, almost closed, ${\theta}-continuous$ and almost quasi-compact mappings in bitopological spaces, Bull. Calcutta Math. Soc. 73 (1981), no. 6, 345-354.
20 J. Dontchev, M. Ganster, and T. Noiri, On p-closed spaces, Int. J. Math. Math. Sci. 24 (2000), no. 3, 203-212.   DOI   ScienceOn
21 H. Z. Hdeib, ${\omega}-continuous$ functions, Dirasat Journal 16 (1989), no. 2, 136-153.
22 D. Jankovic, I. Reilly, and M. Vamanamurthy, On strongly compact topological spaces, Questions Answers Gen. Topology 6 (1988), no. 1, 29-40.
23 M. Jelic, A decomposition of pairwise continuity, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), no. 1, 25-29.
24 J. E. Joseph, Multifunctions and cluster sets, Proc. Amer. Math. Soc. 74 (1979), no. 2, 329-337.   DOI   ScienceOn
25 C. G. Kariofillis, On minimal pairwise Hausdorff bitopological spaces, Indian J. Pure Appl. Math. 19 (1988), no. 8, 751-760.
26 W. J. Thron, Proximity structures and grills, Math. Ann. 206 (1973), 35-62.   DOI
27 A. A. K. Abd El-Aziz, On generalized forms of compactness, Master's thesis, Faculty of Science, Taunta University, Egypt, 1989.
28 M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud, ${\beta}-open$ sets and ${\beta}-continuous$ mapping, Bull. Fac. Sci. Assiut Univ. A 12 (1983), no. 1, 77-90.
29 K. Al-Zoubi and B. Al-Nashef, The topology of ${\omega}-open$ subsets, Al-Manarah Journal 9 (2003), no. 2, 169-179.
30 T. Soundararajan, Weakly Hausdorff spaces and the cardinality of topological spaces, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) pp. 301-306 Academia, Prague, 1971.
31 N. Velicko, H-closed toplogical spaces, Mat. Sb. 70 (1966), 98-112
32 N. Velicko, H-closed toplogical spaces, Amer. Math. Soc. Transl. 78 (1968), no. 2, 103-118.