# Independent form of (?, s)-continuous functions in topological spaces

## DOI:

https://doi.org/10.22199/issn.0717-6279-2020-03-0031## Keywords:

Almost contra-precontinuous functions, (θ, s)-continuous functions, Almost contra-Pβ-continuous functions, Pβ-open, topological space## Abstract

We introduce a new class of almost contra-P?-continuous functions which is a subclass of the class of almost contra-precontinuous functions [8]. This class contains the classes of regular set connected functions, perfectly continuous functions and contra-P?-continuous functions. It is shown that almost contra-P?-continuity is independent to (?, s)-continuity [12] and contra-precontinuity [11]. Furthermore, we obtain basic properties and preservations theorems for almost contra-P?-continuity.## References

M. E. Abd El-Monsef, S. N. El-Deep, and R. A. Mahmoud, “?-open sets and ?-continous mappings”, Bulletin of the Faculty of Science. Assiut University. C, Mathematics, vol. 12, pp. 77-90, 1983.

D. Andrijevi?. "Semi-preopen sets", Matemati?ki Vesnik, vol. 38, no. 93, pp. 24-32, 1986. [On line]. Available: https://bit.ly/2XnaNZ1

S. P. Arya and M. P. Bhamini, “Some generalizations of pairwise urysohn spaces”, Indian journal of pure and applied mathematics, vol. 18, no. 12, pp. 1088-1093, Dec, 1987. [On line]. Available: https://bit.ly/3e1EXYr

C. W. Baker, “On contra almost ?-continuous functions in topological spaces”, Kochi journal of mathematics, vol. 1, pp. 1-8, Mar. 2006.

D. Carnahan, "Some properties related to compactness in topological spaces", PhD thesis, University of Arkansas, 1973.

J. Dontchev, “Contra-continuous functions and stronglyS-closed spaces”, International journal of mathematics and mathematical sciences, vol. 19, no. 2, pp. 303–310, 1996, doi: 10.1155/S0161171296000427

J. Dontchev and M. I. Ganster, “More on almost s-continuity”, Indian journal of mathematics, vol. 41, no. 2, pp. 139-146, 1999.

E. Ekici, “Almost contra-precontinuous functions”, The bulletin of the Malaysian Mathematical Society, vol. 27, no. 1, pp. 53-65, 2004. [On line]. Available: https://bit.ly/2ABhAWZ

E. Ekici, “Another form of contra-continuity”, Kochi journal of mathematics, vol. 1, pp. 21-29, Mar. 2006.

N. Ergun, “On nearly paracompact spaces”, ?stanbul Üniversitesi Fen Fakültesi matematik dergisi, vol. 45, pp. 65-87, 1980.

S. Jafari and T. Noiri, “Contra-precontinuous functions”, The bulletin of the Malaysian Mathematical Society, vol. 25, no. 2, pp. 115-128, 2002. [On line]. Available: https://bit.ly/3cKi1MT

J. E. Joseph and M. H. Kwack, “On S-Closed spaces”, Proceedings of the American Mathematical Society, vol. 80, no. 2, pp. 341–348, Oct. 1980, doi: 10.2307/2042974

N. Levine, “Strong continuity in topological spaces”, The American mathematical monthly, vol. 67, no. 3, p. 269, Mar. 1960, doi: 10.2307/2309695

N. Levine, “Semi-open sets and semi-continuity in topological spaces”, The American mathematical monthly, vol. 70, no. 1, pp. 36–41, Jan. 1963, doi: 10.2307/2312781

O. Njåstad, “On some classes of nearly open sets”, Pacific journal mathematics. vol. 15, no. 3, pp. 961-970, 1965. [On line]. Available: https://bit.ly/3cK1VmB

T. Noiri, “On almost continuous functions”, Indian journal of pure and applied mathematics, vol. 20, no. 6 pp. 571-576, 1989. [On line]. Available: https://bit.ly/2Zgc3iZ

T. Noiri and V. Popa, “A unified theory of almost contra-continuity for functions”, Kochi journal of mathematics, vol. 3, pp. 125-138, Mar. 2008.

G. D. Maio, “S-closed spaces, S-sets and S-continuous functions”, Atti della Accademia delle scienze di Torino. Classe di scienze fisiche matematiche e naturali, vol. 118, pp. 125-134, 1984.

A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, ”On precontinuous and week pre-continuous mappings”, Proceedings of the Mathematical and Physical Society of Egypt, vol. 53, pp. 47-53, 1982.

M. K. Singal, A. R. Singal, and A. Mathur, “On nearly compact spaces”, Bollettino dell'Unione Matematica Italiana, vol. 4, no. 2, pp. 702-710, 1969.

M. K. Singal, A. R. Singal, “A note on mildly compact spaces”, Kyungpook mathematical journal, vol. 19, no. 2, pp. 165-168, Dec. 1979. [On line]. Available: https://bit.ly/3bGZ900

T. Soundarajan, “Weakly Hausdorff space and the cardinality of topological spaces”, in General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur topological conference, 1968, S. P. Franklin, Z. Frolík, and V. Koutník, Eds. Praha: Academia Publishing House of the Czechoslovak Academy of Sciences, 1971, pp. 301–306. [On line]. Available: https://bit.ly/2z2ljN7

L. A. Steen and J. A. Seebach, Counterexamples in topology. New York, NY: Holt, Rinehart und Winston, 1970.

R. Staum, “The algebra of bounded continuous functions into a nonarchimedian field“, Pacific journal of mathematics, vol. 50, no. 1, pp. 169-185, 1974. [On line]. Available: https://bit.ly/3e4iZUm

T. Thompson, “S-closed spaces”, Proceedings of the American Mathematical Society, vol. 60, no. 1, pp. 335–339, Jan. 1976, doi: 10.1090/S0002-9939-1976-0425899-0

B. K. Tyagi, S. Singh, and M. Bhardwaj, “P?-connectedness in topological spaces”, Demonstratio mathematica, vol. 50, no. 1, pp. 299–308, Dec. 2017, doi: 10.1515/dema-2017-0031

B. K. Tyagi, S. Singh, and M. Bhardwaj, “??-connectedness as a characterization of connectedness”, Journal of advanced studies in topology, vol. 9, no. 2, pp. 119-129, 2018. [On line]. Available: https://bit.ly/2XbEPyG

B. K. Tyagi, S. Singh, M. Bhardwaj, and , Harsh V. S. Chauhan, “Some strong forms of connectedness in topological spaces”, Journal of advanced studies in topology, vol. 10, no. 1, pp. 20-27, 2019, doi: 10.20454/jast.2019.1497

B. K. Tyagi, S. Singh, and M. Bhardwaj, “Almost contra-Ps-continuity in topological spaces”, Annales Universitatis Scientiarum Budapestinensis, vol. 62, pp. 61-78, 2019.

B. K. Tyagi, S. Singh, and M. Bhardwaj, “Contra-P?-continuous and contraS?-continuous functions”, Communicated.

G .J. Wang, “On s-closed spaces”, Acta mathematica sinica, vol. 24, no. 1, pp. 55-63, 1981.

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*Proyecciones (Antofagasta, On line)*, vol. 39, no. 3, pp. 495-515, Jun. 2020.

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Copyright (c) 2020 Brij K. Tyagi, Sumit Singh, Manoj Bhardwaj

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