Abstract
We introduce and study \({\mu\mbox{-}S}\)-closed spaces, i.e. \({\mu}\)-spaces \({(X, \mu)}\) in which every cover of X by \({\mu}\)-semi-open sets has a finite subfamily the union of the μ-closures of whose members covers X. The class of \({\mu\mbox{-}S}\)-closed spaces is a proper subclass of \({w\mu}\)-compact spaces.
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References
Abd El-Aziz Ahmed Abo-Khadra, On generalized forms of compactness, Master’s Thesis, Faculty of Science, Tanta University, Egypt (1989).
M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, \({\beta}\)-open sets and \({\beta\mbox{-}}\) continuous mappings, Bull. Fac. Sci. Assiut. Univ., 12 (1983), 77–90.
Andrijević D.: Semi-preopen sets. Mat. Vesnik, 38, 24–32 (1986)
D. Andrijević, On b-open sets, Mat. Vesnik, 48 (1996), 59–64.
D. E. Cameron, Properties of S-closed spaces, Proc. Amer. Math. Soc., 72 (1978), 581–586.
H. H. Corson and E. Michael, Metrizability of certain countable unions, Illinois J. Math., 8 (1964), 351–360.
Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), 351–357.
Á. Császár, Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 53–66.
Á. Császár, Further remarks on the formula for \({\gamma}\)-interior, Acta Math. Hungar., 113 (2006), 325–332.
Á. Császár, Remarks on quasi topologies, Acta Math. Hungar., 119 (2008), 197–200.
Di Maio and T. Noiri, On s-closed spaces, Indian J. Pure Appl. Math., 18 (1987), 226–233.
J. Dontchev, M. Ganster and T. Noiri, On p-closed spaces, Internat. J. Math. Math. Sci., 24 (2000), 203–212.
R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics 6, Heldermann Verlag (Berlin, 1989).
Levine N.: Semi-open sets and semi-continuity in topological spaces. Amer. Math. Monthly, 70, 36–41 (1963)
A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53.
O. Njåstad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970.
T. Noiri, On S-closed subspaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8) (1978), 157–162.
T. Noiri, Properties of S-closed spaces, Acta Math. Acad. Sci. Hungar., 35 (1980), 431–436.
T. Noiri, Unified characterizations for modifications of \({R_{0}}\) and \({R_{1}}\) topological spaces, Rend. Circ. Mat. Palermo (2), 55 (2006), 29–42.
J. R. Porter and J. D. Thomas, On H-closed and minimal Hausdorff spaces, Trans. Amer. Math. Soc., 138 (1969), 159–170.
M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar., 131 (2011), 110–121.
Sarsak M. S.: B-closed spaces. Demonstratio Math., 45, 195–206 (2012)
M. S. Sarsak, Weakly \({\mu}\)-compact spaces, Demonstratio Math., 45 (2012), 929–938.
M. S. Sarsak, On \({\mu}\)-compact sets in \({\mu}\)-spaces, Questions Answers Gen. Topology, 31 (2013), 49–57.
M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstratio Math., 46 (2013), 415–427.
T. Thompson, S-closed spaces, Proc. Amer. Math. Soc., 60 (1976), 335–338.
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Sarsak, M.S. \({\mu\mbox{-}S}\)-closed spaces. Acta Math. Hungar. 146, 285–299 (2015). https://doi.org/10.1007/s10474-015-0532-7
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DOI: https://doi.org/10.1007/s10474-015-0532-7
Key words and phrases
- \({\mu}\)-opens
- \({\mu}\)-closed
- \({\mu}\)-semi-open
- \({\mu}\)-regular open
- \({\mu}\)-regular closed
- \({\mu}\)-semi-regular
- generalized topology
- \({\mu}\)-space
- \({w\mu}\)-compact set
- \({w\mu}\)-compact space
- \({\mu\mbox{-}S}\)-closed set
- \({\mu\mbox{-}S}\)-closed space