Document Type: Research Paper


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

2 College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark

3 Departamento de Mathematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n24020-140, Niteroi, RJ Brasil

4 Department of Mathematics and statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

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


In this paper, we show that a pointwise symmetric pre-isotonic preclosure function is uniquely determined the pairs of sets it separates. We then show that when the preclosure function of the domain is pre-isotonic and the pre-closure function of the codomain is pre-isotonic and pointwise-pre-symmetric, functions which separate only those pairs of sets which are already separated are precontinuous.


Main Subjects

[1] M. Caldas, S. Jafari, R. M. Latif, A. A. Nasef, Semi-continuity and semi-connectedness in generalized semiclosure spaces, King Fahd University of Petroleum and Minerals, Dep. of Math. Sci. (386) (2008), 1-13.

[2] H. Corson, E. Michael, Metrizability of cerain countable uxions, Illinois J. Math. 8 (1964), 351-360.

[3] S. N. El-Deeb, I. A. Hasanein, A. S. Mashhour, T. Noiri, On p-regular spaces, Bull Math. Soc. Sci. R. S. Roumaine. 27 (57) (1983), 311-315.

[4] M. S. El-Naschie, On the uncertainty of Cantorian geometry and two slit experiment, Chaos. Solitons and Fractals. 9 (3) (1998), 517-529.

[5] E. D. Kalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topol. Appl. 36 (1990), 1-17.

[6] A. Kar, P. Bhattacharyya, Some weak Separation axioms, Bull. Calcutta Math. Soc. 82 (1990), 415-422.

[7] A. 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.

[8] V. Ptak, Completeness and the open mapping theorem, Bull. Soc. Math. France. 86 (1958), 41-74.

[9] M. B. Smyth, Semi-Matrices, Closure Spaces and digital topology, Theore. Comput. Sci. (5) (1995), 257-276.