A Unified Theory of Weakly ${G}$-Closed Sets and Weakly ${G}$-Continuous Functions
DOI:
https://doi.org/10.5644/SJM.09.1.12Keywords:
$m$-structure, weakly $g$-closed, weakly $\omega$-closed, weakly $rg$-closed, weakly $\pi g$-closed, $wmng$-closed set, $wmng$-continuousAbstract
We introduce the notion of weakly $mng$-closed sets as a unified form of weakly $\omega$-closed sets \cite{Ra}, weakly $rg$-closed sets \cite{Na}, weakly $\pi g$-closed sets \cite{RGC} and weakly $mg^\ast$-closed sets \cite{No-Po4}. Moreover, we introduce and study the notion of weakly $mng$-continuous functions to unify some modifications of weakly $g$-continuous functions.
2010 Mathematics Subject Classification. 54A05, 54C08
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M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, β-open sets and βcontinuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77–90.
M. E. Abd El-Monsef, R. A. Mahmoud and E. R. Lashin, $beta$-closure and$beta$-interior, J. Fac. Ed. Ain Shams Univ., 10 (1986), 235–245.
D. Andrijevi´c, Semi-preopen sets, Mat. Vesnik, 38 (1986), 24–32.
D. Andrijevi´c, On b-open sets, Mat. Vesnik, 48 (1996), 59–64.
K. Balachandran, P. Sundarm and H. Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 12 (1991), 5–13.
C. Boonpok, Almost and weakly M-continuous functions in m-spaces, Far East J. Math. Sci., 43 (2010), 29–40.
C. Boonpok, Biminimal structure spaces, Int. Math. Forum, 5 (15) (2010), 703–707.
M. Caldas, On g-closed sets and g-continuous mappings, Kyungpook Math. J., 33 (1993), 205–209.
M. Caldas, Further results on generalized open mappings in topological spaces, Bull. Calcutta Math. Soc., 88 (1996), 37–42.
M. Caldas, S. Jafari and T. Noiri, Notions via g-open sets, Kochi J. Math., 2 (2007), 43–50.
S. G. Crossley and S. K. Hildebrand, Semi-closure, Texas J. Sci., 22 (1971), 99–112.
J. Dontchev and T. Noiri, Quasi normal spaces and πg-closed sets, Acta Math. Hungar., 89 (2000), 211–219.
W. Dunham and N. Levine, Further results of genralized closed sets in topology, Kyungpook Math. J., 20 (1980), 169–175.
S. N. El-Deeb, I. A. Hasanein, A. S. Mashhour and T. Noiri, On p-regular spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 27 (75) (1983), 311–315.
N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41.
N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo (2), 19 (1970), 89–96.
H. Maki, K. C. Rao and A. Nagoor Gani, On generalizing semi-open and preopen sets, Pure Appl. Math. Sci., 49 (1999), 17–29.
A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deep, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53.
A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, α-continuous and α-open mappings, Acta Math. Hungar., 41 (1983), 213–218.
W. K. Min and Y. K. Kim, $M^ast$-continuity and product minimal structure on minimal structures, Int. J. Pure Appl. Math., 69 (3) (2011), 329–339.
B. M. Munshi and D. S. Bassan, g-continuous mappings, Vidya J. Gujarat Univ. B Sci., 24 (1981), 63–68.
M. Murugalingam, A Study of Semi-generalized Topology, Ph. D. Thesis, Manonmaniam Sundaranar Univ., Tamil Nadu (India), 2005.
N. Nagoveni, Studies of Generalizations of Homeomorphisms in Topological Spaces, Ph. D. Thesis, Bharathiar Univ., Coimbatore (India), 1999.
O. Nj˚astad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970.
T. Noiri, The further unified theory for modifications of g-closed sets, Rend. Circ. Mat. Palermo, 57 (2008), 411-421.
T. Noiri and V. Popa, Between closed sets and g-closed sets, Rend. Circ. Mat. Palermo (2), 55 (2006), 175–184.
T. Noiri and V. Popa, A unified theory of weak continuity for multifunctions, Stud. Cerc. St. Ser. Mat., Univ. Bacˇau, 16 (2006), 167–200.
T. Noiri and V. Popa, A generalization of ω-continuity, Fasciculi Math., 45 (2010), 71–86.
T. Noiri and V. Popa, A generalization of $omega^ast$-continuity, Math. Macedonica (to appear).
N. Palaniappan and K. C. Rao, Regular generalized closed sets, Kyungpook Math. J., 33 (1993), 211–219.
R. Parimelazhagan, K. Balachandran and N. Nagaveni, Weakly generalized closed sets in minimal structure, Int. J. Contemp. Math. Sci., 4 (27) (2009), 1335–1343.
V. Popa and T. Noiri, On M-continuous functions, Anal. Univ. ”Dunˇarea de Jos” Galat¸i, Ser. Mat. Fiz. Mec. Teor. (2), 18 (23) (2000), 31–41.
V. Popa and T. Noiri, On the definitions of some genralized forms of continuity under minimal conditions, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 22 (2001), 9–18.
V. Popa and T. Noiri, On the points of continuity and discontinuity, Bull. U. P. G. Ploesti, Ser. Mat. Fiz. Inform., 53 (2001), 95–100.
V. Popa and T. Noiri, A unified theory of weak continuity for functions, Rend. Circ. Mat. Palermo (2), 51 (2002), 439–464.
V. Popa and T. Noiri, On almost m-continuous functions, Math. Notae, 40 (1999-2002), 75–94.
V. Popa and T. Noiri, On weakly m-continuous functions, Mathematica (Cluj), 45 (68) (2003), 53–67.
N. Rajesh, On weakly ω-closed sets in topological spaces, Math. Macedonica, 3 (2005), 15–24.
K. C. Rao and K. Joseph, Semi-star generalized closed sets, Bull. Pure Appl. Sci., 19 E (2) (2000), 281–290
O. Ravi, S. Ganesan and S. Chandraseker, On weakly g-closed sets, (submitted).
P. Sundaram and M. Sheik John, Weakly closed sets and weakly continuous maps in topological spaces, Proc. 82nd Indian Science Congress, Calcutta, 1995, p. 49.
P. Sundram and N. Nagoveni, On weakly generalized continuous maps,weakly generalized closed maps and weakly generalized irresolute maps in topological spaces, Far East J. Math. Sci., 6 (6) (1998), 903–912.
M. K. R. S. Veera Kumar, On gˆ-closed sets in topological spaces, Bull. Allahabad Math. Soc., 18 (2003), 99–112.
N. V. Veliˇcko, H-closed topological spaces, Amer. Math. Soc. Transl. (2), 78 (1968), 103–118.
V. Zaitsev, On certain classes of topological spaces and their bicompactifications, Dokl. Acad. Nauk SSSR, 178 (1968), 778–779.
