Document Type: Research Paper


1 Department of Physics and Engineering Mathematics , Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt

2 Department of Mathematics, Faculty of Science, Assuit University, New Valley, Egypt


In this paper, we define and study two operators $\Phi^s$ and $\Psi^s$ with grill. Characterization and basic properties of these operators are obtained. Also, we generalize a grill topological spaces via topology $\tau^s$ induced from operators $\Phi^s$ and $\Psi^s$.


Main Subjects

[1] A. Al-Omari, T. Noiri, Decomposition of continuity via grilles. Jordan J. Math and Stat., 4 (1) (2011), 33-46.

[2] C. W. Baker, Slightly precontinuous functions. Acta Math. Hungar., 94 (1-2)(2002).

[3] G. Choquet, Sur les notions de lter et. grill. Completes Rendus Acad. Sci. Paris, 224 (1947), 171-173.

[4] K. C. Chattopadhyay, O. Njastad and W. J. Thron, Merotopic spaces and extensions of closure spaces. Can. J. Math., 35 (4) (1983), 613-629.

[5] K. C. Chattopadhyay and W. J. Thron, Extensions of closur spaces. Can. j. Math., 29 (6) (1977), 1277-1286.

[6] S. G. Crossely and S. K. Hildebrand, Semi-closure. Texas J. Sci. (2+3) (1971), 99-119.

[7] E. Hater and S. Jafari, On Some new Classes of Sets and a new decomposition of continuity via grills. J. Adv. Math. Studies, 3(1) (2010), 33-40.

[8] R. C. Jain, The Role of Regularly Open Sets in General Topology, Ph.D. Thesis. Meerut Univ., (Meerut, India, 1980).

[9] D. Mondal and M.N. Mukherjee, On a class of sets via grill: A decomposition of continuity. An. St. Univ. Ovidius Constanta, 20 (1) (2012), 307-316.

[10] N. Levine, Semi-open sets and semi-continuity in topological spaces. Amer, Math. Mounthly, 86 (1961), 44-46.

[11] B. Roy and M. N. Mukherjee, On a Typical Topology Induced by a Grill. Soochow Journal of Math., 33, No. 4 (2007), 771-786.

[12] R. Staum. The algebra of bounded continuous functions into a nonarchimedian eld. Paci c J. Math., 50 (1974), 169-185.

[13] W. J. Thron, Proximity structure and grills. Math. Ann. 206 (1973), 35-62.