# Characterizations of upper and lower $\alpha(\mu_X,\mu_Y)$-continuous multifunctions

Volume 17, Issue 2, pp 255-265

Publication Date: 2017-06-15

http://dx.doi.org/10.22436/jmcs.017.02.07

### Authors

Napassanan Srisarakham - Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, Thailand.
Chawalit Boonpok - Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, Thailand.

### Abstract

A new class of multifunctions, called upper (lower) $\alpha(\mu_X,\mu_Y)$-continuous multifunctions, has been defined and studied. Some characterizations and several properties concerning upper (lower) $\alpha(\mu_X,\mu_Y)$-continuous multifunctions are obtained.

### Keywords

Generalized topological space, $\mu-\alpha$-open, upper $\alpha(\mu_X،\mu_Y)$-continuous multifunction, lower $\alpha(\mu_X،\mu_Y)$-continuous multifunction.

### References

[1] M. E. Abd El-Monsef, S. N. El-Deeb, R. A. Mahmoud, $\beta$-open sets and $\beta$-continuous mapping, Bull. Fac. Sci. Assiut Univ. A, 12 (1983), 77–90.
[2] M. E. Abd El-Monsef, A. A. Nasef, On multifunctions, Chaos Solitons Fractals, 12 (2011), 2387–2394.
[3] D. Andrijevi´c, Semi-preopen sets, Math. Vesnik, 38 (1986), 24–32.
[4] C. Boonpok, On upper and lower $\beta(\mu_X,\mu_Y)$-continuous multifunctions, Int. J. Math. Math. Sci., 2012 (2012), 17 pages.
[5] A´ . Csa´sza´r, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), 351–357.
[6] A´ . Csa´sza´r, -connected sets, Acta Math. Hungar., 101 (2003), 273–279.
[7] A´ . Csa´sza´r, Extremally disconnected generalized topologies, Ann. Univ. Sci. Budapest. Eo¨ tvo¨ s Sect. Math., 47 (2004), 91–96.
[8] A´ . Csa´sza´r, Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 53–66.
[9] A´ . Csa´sza´r, Further remarks on the formula for $\gamma$-interior, Acta Math. Hungar., 113 (2006), 325–332.
[10] A´ . Csa´sza´r, Modification of generalized topologies via hereditary classes, Acta Math. Hungar., 115 (2007), 29–36.
[11] A´ . Csa´sza´r, $\delta$- and $\theta$-modifications of generalized topologies, Acta Math. Hungar., 120 (2008), 275–279.
[12] A´ . Csa´sza´r, Product of generalized topologies, Acta Math. Hungar., 123 (2009), 127–132.
[13] A. Deb Ray, R. Bhowmick, $\mu$-paracompact and $g_\mu$-paracompact generalized topological spaces, Hacet. J. Math. Stat., 45 (2016), 447–453.
[14] A. Kanibir, I. L. Reilly, Generalized continuity for multifunctions, Acta Math. Hungar., 122 (2009), 283–292.
[15] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41.
[16] M. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53.
[17] T. Neubrunn, Strongly quasi-continuous multivalued mappings, General topology and its relations to modern analysis and algebra, VI, Prague, (1986), 351–359, Res. Exp. Math., Heldermann, Berlin, (1988).
[18] O. Nja˙ stad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970.
[19] T. Noiri, V. Popa, Almost weakly continuous multifunctions, Demonstratio Math., 26 (1993), 363–380.
[20] T. Noiri, V. Popa, On upper and lower almost $\beta$-continuous multifunctions, Acta Math. Hungar., 82 (1999), 57–73.
[21] J. H. Park, B. Y. Lee, M. J. Son, On upper and lower $\delta$-precontinuous multifunctions, Chaos Solitons Fractals, 19 (2004), 1231–1237.
[22] V. I. Ponomarev, Properties of topological spaces preserved under multivalued continuous mapping on compacta, Amer. Math. Soc. Translations, 38 (1964), 441–446.
[23] V. Popa, Sur certaines formes faibles de continuit´e pour les multifonctions, (French) [[Some weak forms of continuity for multifunctions]] Rev. Roumaine Math. Pures Appl., 30 (1985), 539–546.
[24] V. Popa, Some properties of $H$-almost continuous multifunctions, Problemy Mat., 10 (1990), 9–26.
[25] V. Popa, T. Noiri, On upper and lower $\alpha$-continuous multifunctions, Math. Slovaca, 43 (1993), 477–491.
[26] V. Popa, T. Noiri, On upper and lower almost $\alpha$-continuous multifunctions, Demonstratio Math., 29 (1996), 381–396.
[27] V. Popa, T. Noiri, On upper and lower $\beta$-continuous multifunctions, Real Anal. Exchange, 22 (199697), 362–376.
[28] V. Popa, T. Noiri, On upper and lower weakly $\alpha$-continuous multifunctions, Novi Sad J. Math., 32 (2002), 7–24.
[29] B. Roy, On a type of generalized open sets, Appl. Gen. Topol., 12 (2011), 163–173.
[30] M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar., 131 (2011), 110–121.
[31] R.-X. Shen, A note on generalized connectedness, Acta Math. Hungar., 122 (2009), 231–235.
[32] R.-X. Shen, Remarks on products of generalized topologies, Acta Math. Hungar., 124 (2009), 363–369.