# A Note on Strongly Lower Semi-Continuous Functions and the Induced Fuzzy Topological Space Generated by Them

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2013.45.1.6## Keywords:

Topology, fuzzy topological space, SLSC function, regular F(sigma)-subset, s-induced fuzzy topological space.## Abstract

A new class of functions called strongly lower semi-continuous (SLSC) functions is defined and its properties are studied. It is shown that the arbitrary supremum and finite infimum of SLSC functions are again SLSC. Using these functions, an induced fuzzy topological space, called s-induced fuzzy topological space on a topological space (X, T), is introduced. Moreover, some incorrect results on fuzzy topological spaces obtained previously by some authors are identified and modified accordingly. Examples of the newly defined induced space are given and their various properties are investigated. Interrelationships between a fuzzy topological space (X,*F*) and the s-induced fuzzy topological space generated by the crisp members of

*F*are examined. In this process, different lower semi-continuities and induced fuzzy spaces generated by them have been defined in a general set up and their few properties have been studied.

## References

Chang, C.L., Fuzzy Topological Spaces, J. Math. Anal. Appl., 24, pp. 182-190, 1968.

L.A. Zadeh, Fuzzy Sets, Inform. and control, 8, pp. 338-353, 1965.

R. Lowen, Fuzzy Topological Spaces and Fuzzy Compactness, J. Math. Anal. Appl. 56, pp. 621-633, 1976.

M.D. Weiss, Fixed Points, Separation, and Induced Topologies for Fuzzy Sets, J. Math. Anal. Appl., 50, pp. 142-150, 1975.

Bhaumik, R.N. & Mukherjee, A., Completely Lower Semi-Continuous Functions, Math. Edu. 26, pp. 66-69, 1992.

Mukherjee, A. & Halder, S., -Induced Fuzzy Topological Spaces, Proc. Nat. Sem. on Recent Trends in Maths. & Its Appl., April 28-29, 2003, pp. 177-182, 2003.

Bhaumik, R.N. & Bhattacharya, D., On Regular Semi-Continuous Functions, Math. Edu., xxvi(1), pp. 11-17, 1992.

Bhaumik, R.N. & Mukherjee, A., Some More Results on Completely Induced Fuzzy Topological Spaces, Fuzzy Sets and Systems, 50, pp. 113-117, 1992.

Monsef, M.E. Abd. El. & Ramadan, A.E., On Fuzzy Supra Topological Spaces, Indian J. Pure and Appl. Math., 18, pp. 322-329, 1987.

Mukherjee, A., Some More Results on Induced Fuzzy Topological Spaces, Fuzzy Sets and Systems, 96, pp. 255-258, 1998.

Bhaumik, R.N. & Bhattacharya, D., Regular G-Subsets and Real Compact Spaces, Bull. Cal. Math. Soc., 87, pp. 39-44, 1995.

Dilworth, R.P., The Normal Completion of Lattice of Continuous Functions, Trans. Amer. Math. Soc., 68, pp. 427-438, 1950.

Bhattacharjee, D. & Saha, A.K., Fuzzy Topological Spaces Induced by Regular Lower Semi-Continuous Functions, Proc. Nat. Sem. on Fuzzy Math. & Its Appl., Nov. 25-26, pp. 47-56, 2006.

Saha, A.K. & Bhattacharya, D., A Study on Induced Fuzzy Topological Space Generated By M-RLSC Functions, Proceedings of International Conference on Rough Sets, Fuzzy Sets and Soft Computing, Nov. 5-7, pp. 400-408, 2009.

Alimohammady, M. & Roohi, M., Fuzzy Minimal Structure and Fuzzy Minimal Vector Spaces, Chaos, Solitons and Fractals, 27, pp. 599-605, 2006.

El Naschie, M.S., On The Uncertainty of Cantorian Geometry and The Two-Slit Experiment, Chaos, Solitons and Fractals, 9(3), pp. 517-529, 1998.

El Naschie, M.S., On The Unification of Heterotic Strings, M Theory and E(a??) Theory, Chaos, Solitons and Fractals, 11(14), pp. 2397-2408, 2000.

Kelley, J.L., General Topology, D. Van Nostrand, Princeton, NJ, 1955.

Bhaumik, R.N. & Mukherjee, A., Completely Induced Fuzzy Topological Spaces, Fuzzy Sets and Systems, 47, pp. 387-390, 1992.

Gillman, L. & Jerrison, M., Rings of Continuous Function, Van Nostrand, 1960.

Mack, J.E., Countable Paracompactness and Weak Normality Properties, Trans. Amer. Math. Soc., 148, pp. 256-272, 1970.

Palaniappan, N., Fuzzy Topology, Alpha Science International Ltd; 2002.

Rodabaugh, S.E., Powerset Operator Based Foundation for Point-Set Lattice-Theoretic (Poslat) Fuzzy Set Theories and Topologies, Quaest. Math., 20(3), pp. 463-530, 1997.

Geping, W. & Lanfang, H., On Induced Fuzzy Topological Space, J. Math. Anal. Appl., 108, pp. 495-506, 1985.

Bhattacharya, D. & Saha, A.K., A Note On R-Countably Induced Fuzzy Topological Space, Proc. Nat. Sem. On Rec. Dev. in Math. & Its Appl. Nov. 14-15, pp. 1-5, 2008.