Abstract:

The main purpose of the present work is to introduce and study a new class of semi-open sets called -open sets. We use this set to study new topological concepts such as -continuous functions, almost and weakly -continuous functions, contra -continuous functions, -compact and -connected spaces. Eventually, new types of -separation axioms are also studied.

At the beginning several properties of -open sets are obtained. It is proved that if a topological space  is semi- , then the family of semi-open sets is equal to the family of -open sets. Moreover, if a topological space  is semi-regular, then every open set is -open set. It is shown that for any space, every singleton subset of  is -open if and only if it is semi-regular.

For -continuous functions, almost and weakly -continuous functions several results and characterizations are obtained. It is proved that the restriction of -continuous to -open, semi-closed set is also -continuous. Moreover, it is shown that if the co-domain  is hyperconnected, then for any function  is almost -continuous. Furthermore, we determine the relation between almost -continuous and weakly -continuous functions as follows for a function  and  is extremely disconnected, then  is almost -continuous if and only if it is weakly -continuous.

The notions of -compact spaces and -sets are studied. It is showed that -compactness is strong form of semi-compactness. Also, if  is -compact, then it is quasi H-closed and hence it is mildy compact. We prove that a space  is -compact if and only if every proper -closed subset of  is -set. Regarding to -connected space, we prove that a topological space  is -connected if and only if the -closure of every non-empty -open set is. Finally we define generalized forms of semi-separation axioms are called -separation axioms where several characterizations are given and the relations among them are found.

References

  • Abdul-Jabbar A. M., (2000), “ s-continuity, openness and closed graphs in topological spaces”, M. Sc. Thesis, College of Science, Salahaddin University.
  • Ahmed N.K., (1990), “On Some Types Of Separation Axioms”, M.Sc. Thesis, College Of Science, Salahaddin University .
  • Al-Nashef , (2000), “Countably I-compact spaces”, Hindawi publishing corp., pp.745-751.
  • Andrivic D., (1984), “ Some properties of the topology of -sets”, Math. Vesnik, Vol.36, pp.1-10.
  • Arya S. P. and Bhamini M. P., (1996), “Some weaker forms of semi-continuous functions”, Ganita , Vol.33, No.2, pp.124-134.
  • Balachandran K., Sundaram P., and Maki H., (1999), “On generalized continuous maps in topological spaces”, Fac. Sci. Koch Univ. Ser. A(Math.), Vol.12, pp. 5-13.
  • Bhattacharya P. and Lahiri B.K., (1985), “Semi-generalized closed sets in topology”, Indian J. Pure Appl. Math, 29, No.3, pp. 375-382.
  • Caldas M., Jafari S., Noiri T. and Saraf R., (2010),“Functions with strongly semi- -closed graphs”, of advanced Research in Applied Mathematics, Vol.2, No.2, pp. 38-47.
  • Carnahan D. A., (1973), “Some properties related to compactness in topological spaces”, Ph.D. Thesis, University of Arkansas.
  • Crossely S. G. and Hildebrand S. K., (1971), “Semi closure”, Texas J. Sci., Vol.22, pp.99-112.
  • Crossely S. G. and Hildebrand S. K., (1972), “Semi-topological  properties”, Fundamenta Methematicae, pp.233-254.
  • Das P.,(1974), “Note on semi-connectedness” Indian J. of Mechanics and Mathematics, Vol.12, pp.31-35.
  • Di Maio G. and Noiri T., (1987), “On -closed spaces, Indian J. Pure Appl. Math., Vol.18, No. 3, pp.226-233.
  • Dlaska K., Ergun N. and Ganster M, (1994),“Countably S-closed spaces”, Slovaca, Vol.44, No.3, pp.337-348.
  • Dlaska K., Ergun N. and Ganster M., (1995), “ On the topology generated by semi-regular sets”, Indian J. Pure Appl. Math., Vol.25, No.11, pp.1163-1170.
  • Dontchev J., (1996), “Contra-continuous functions and strongly S-closed spaces”, internat J. Math. & Math. Sci., Vol.19, No.2, pp.303-310.
  • Dontchev J., (1998), “Survey on pre-open sets”, Ar Xiv: Math., Vol. 1, pp.1-18.
  • Dontchev J. and Ganster M., (1996), “On covering spaces with semi-regular sets”, Ricerche di Matematica, Vol.45, No.1, pp.229-245.
  • Dontchev J., Ganster M. and Noiri T., (2000), “On P-closed spaces”, internat J. Math. & Math. Sci., 24(3), pp.203-212.
  • Dontchev J. and Noiri T., (1999), “Contra semi-continuous functions”, Mathemtica Pannonica, Vol.10, No.2, pp. 159-168.
  • Dorsett C., (1981), “Semi-compactness, semi separation axioms and product spaces”, Malaysian Math.Soc., Vol.2, No.4, pp.21-29.
  • Dorsett C., (1982), “Semi convergence and semi-compactness”, Indian J. Mech. Math., 19, No.1, pp.11-17.
  • Dorsett C., (1978), “Semi- , Semi- , and Semi- topological spaces”, di. Bruxelles,T. Vol.92,No.3, pp. 143-150.
  • Dorsett C., (1982), “Semi-regular spaces”, Soochow J. Math., 8, pp. 45-53.
  • Ekici E., (2008), “On contra -continuous functions in topological space”, Choas, solitons and fractlas, Vol.35, pp.71-81.
  • Herrmann R. A., (1987), “rc-convergence”, Amer. Math. Soc., Vol.75, No.2, pp.311-317.
  • Jankovic D. S. and Rielly I. L., (1985), “On semi-separation properties”, Indian J. Pure Appl. Math., Vol.16, No.9, pp.957-964.
  • Joseph J. E. and Kwack M. H., (1980), “On -closed spaces”, Amer. Math. Soc., Vol.80 No.2, pp. 341-348.
  • Khalaf A. B. and Ameen Z. H., (2010), “ -open sets and -continuous functions”, of Advanced Research in Pure Mathematics, Vol.2, No.3, pp.87-101.
  • Khalaf A. B. and Easif F. H., (1999), “ -continuous functions”, Dohuk Univ., (special issue), Vol.2, No.1, pp.1-7.
  • Kohli J. K. and Das A. K., (2002), “New normality axioms and decomposition of normality”, Mate.Vol.37, No.57, pp. 165-175.
  • Levine N., (1970), “Generalized closed in topology”, Circlo. Mat. Palermo, Vol.19, No.2, pp.89-96.
  • Levine N., (1963), “Semi-open sets and semi-continuity in topological spaces”, Math. Monthly, Vol.70, pp. 36-41.
  • Long P. E. and Herrington L., (1981), “ Strongly -continuous functions”, Korean. Math. Soc.18, pp. 21-28.
  • Maheshwari S. N. and Prasad R., (1975), “ Some new separation axioms”, Soc. Sci. Bruxelles, ser. I., Vol.89, pp. 395-402.
  • Mashhour A.S. and Abd-El-monsef M.E. and El-Deeb S.N., (1982), “ On pre-continuous and weak pre-continuous mapping”, Amer. Phys. Soc. Egypt, Vol.53, pp. 47-53.
  • Mathur A., “A note on S-closed spaces”, Amer. Math. Soc, Vol.174, No.2, pp.350-352.
  • Mirmiran M., (2000), “A survey on extremally disconnected spaces”, Math. Dep., Isfahan Univ., Iran, pp.1-3.
  • Njastad O.,(1965), “On some classes of nearly open sets”, Pacific J. Math., Vol.15, pp.961-970.
  • Noiri T.,(1978),“A generalization of perfect functions”, London Math.Soc., 2(17)(1978),pp.540-544.
  • Noiri T.,(1989), “On almost continuous functions”, Indian J. Pure Appl. Math., Vol.20, No.6, pp.571-576.
  • Noiri T., (1984), “Supercontinuity and some strong forms of continuity”, Indian J. Pure Appl. Math., Vol.15, No.3, pp.241-250.
  • Park J. H., Beom Y., and Lee B. Y., (2002), “On gp-closed and pre gp-continuous functions”, Indian J. Pure Appl. Math., Vol.33, No.1, pp.3-12.
  • Pipitone V., and Russo G., (1975), “Spazi semiconnessi e spazi semiaperti”, Rend Circ. Mat. Palermo, 24(2), pp. 273-285.
  • Porter J. R. and Woods R. G., (1984), “Feebly compact spaces” , Martin’s axioms and diamond, Topology Proc.9(1984), pp.105-121.
  • Ptak V., (1958), “Completeness and open mapping theorem”, Soc. Math.France, Vol.86, pp. 41-74.
  • Puturong N., (2007), “On strongly -semi-continuous functions”, Thai Journal of Mathematics, special Issue Annual meeting in mathematics, pp.11-23.
  • Saleh M., (2004), “On -closed sets and some forms of continuity”, Archivum Mathemticum, 40, pp.383-393.
  • Sarker J. P., and Dasgupta H., (1985), “Locally semi-connected in topological spaces”, Indian J. Pure Appl. Math., Vol. 16(12), pp. 1488-1494.
  • Shanin N. A., (1943), “On separation in topological space”, R. (Doklady) Acad. Sci. URSS (N.S.), Vol.38, pp.110-113.
  • Sharma J. N., (2011), “Topology (General and Algebraic)”, Krishna Parkashan Mandir, 4th
  • Singal M. K. and Singal A. R., (1968), “Almost continuous mappings”, Yokohama Math. J., Vol. 16, pp.63-73.
  • Staum R., (1974), “the algebra of bounded continuous functions into non archimedean field”, Pacific J. Math.,50, pp.169-185.
  • Steen L.A., Seebach Jr. J.A., (1970), “Counter examples in topology”, Holt, Rinehart and Winston, Inc., New York.
  • Tadros S. F. and Khalaf A. B., (1992), “On regular semi-open sets and S*-closed spaces”, Tamkang J. of Math., Vol.23, No.4, pp. 337-348.
  • Thompson T., (1976), “S-closed spaces”, Amer. Math. Soc.,Vol.60, No.1, pp.335-338.
  • Tong J.,(1989), “ On decomposition of continuity in topological spaces”, Math. Hungar, Vol.54, No.(1-2), pp. 51-55.
  • Velico N. V., (1968), “H-closed topological spaces” , Math. Soc. Trans., Vol. 2, pp. 103-118.
  • Yang C. T., (1954), “On paracompact spaces”, Amer. Math. Soc., Vol.5, pp. 185-189.

 

 

 

 

€120.00
Purchase