**Abstract:**

Introducing and investigating a new class of sets called Pp -open sets is the main aim of this work and using this set we define and study new classes of the most well- known notions topological space such as continuity and new types of separation axioms.

We start this work by defining the class of Pp -open sets which is contained in the class of preopen sets and contains the class of pre-θ-open sets. It is shown that the family of Pp O(X ) sets form a topology on a space X , if X is locally indiscrete or has a property P . It is also appeared that the family of Pp O(X ) sets is discrete topology in X , if (X, τ ) is indiscrete topology. It has also been proved that the family of Pp -open sets and preopen sets are identical, if X is pre-T1 or locally indiscrete spaces. It is shown that if X is pre-regular, then τ ⊆ Pp O(X ) and for any subset A of a space X , we have Pp Bd [Pp Bd{Pp Bd(A)}] = Pp Bd [Pp Bd(A)]. In the last section, the Pp -g.closed sets have been defined and characterization theorems are provided in some spaces.

Moreover, some new types of separation axioms are provided and defined. It is

proved that Pp -T1 gives Pp -T 1 2 and Pp -T 1 2 gives Pp -T0 . It is shown that a space

X is Pp -T 1 2 if and only if every singleton is either Pp -closed or Pp -open. It has

also been proven that the notions of Pp -D1 and Pp -D2 are equivalent. In addition, the properties and characterizations of Pp -R0 , Pp -R1 and Pp -regular spaces are studied.

Finally, the concepts of Pp -continuous and weakly Pp -continuous are defined. It is proved that Pp -continuity is weaker than strongly θ-continuous, but it is stronger than precontinuity. It is observed that weakly Pp -continuity is weaker than Pp – continuity and stronger than weakly precontinuity. Furthermore, many conditions are provided that make Pp -continuity and weakly Pp -continuity equivalent to some other types of continuity. It is demonstrated that the inverse image of each D- set in Y is a Pp D-set in X , if f : X → Y is a Pp -continuous surgective function. Moreover, it is perceived that X is a Pp -D1 space, if Y is a D-set and f : X → Y is a Pp -continuous bijective function. It has been proven that there are relationships between X and Y with respect to separation axioms, if f : X → Y is a Pp – continuous.

Bibliography

[1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. Assuit. Univ., 12 (1) (1983), 1-18.

[2] N. K. Ahmed, On some types of separation axioms, M.Sc. Thesis, College of Science, Salahaddin Univ., 1990.

[3] D. Andrijevic, Some properties of the topology of α-sets, Math. Vesnik, 36 (1984), 1-10.

[4] C. W. Baker, Weakly θ-precontinuous functions, Acta Math. Hungar. 100 (2003), 342-351.

[5] M. Caldas, S. Jafari and T.Noiri, Characterizations of Pre-R0 and Pre-R1 Topological Spaces, Topology Proceedings, Vol. 25, Summer 2000, 17-30.

[6] S. H. Cho, A note on strongly θ-precontinuous functions, Acta Math. Hungar., 101 (1-2)(2003), 173-178.

[7] J. Dontchev, Survey on preopen sets, The Proceedings of the Yatsushiro Topo- logical Conference, (1998), 1-18.

[8] S. N. El-Deeb, I. A. Hasanein, A. S. Mashhour and T. Nori, On P-regular spaces, Bull. Math. Soc. Sci. Math. R.S. Roum, 27 (4) (1983), 311-315.

[9] S. Jafari, On a weak separation axiom, Far East J. Math. Sci., 3 (5) (2001), 779-787.

[10] J. E. Joseph and M. H. Kwack, On S-closed spaces, Proc. Amer. Math. Soc., 80 (2) (1980), 341-348.

[11] A. Kar and P. Bhattacharyya, Some weak separation axioms, Bull. Cal. Math. Soc., 82 (1990), 415-422.

[12] G. Di Maio, On semi topological operator and semi separation axioms, Rend. Circ. Math. Palermo (2) Suppl. Second Topology Conference, 12 (1986), 219-230.

[13] N. Levine, A decomposition of continuity in topological space, Amer. Math. Monthly 68 (1961), 44-46.

[14] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1) (1963), 36-41.

[15] P. E. Long and L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc., 18 (1) 1981, 21-28.

[16] H. Maki, J. Umehara and T. Noiri, Every topological space is Pre-T 1 , Mem. 2 Fac. Sci. Kochi Univ. Ser. A Math., 17 (1996), 33-42.

[17] A. S. Mashhour, M. E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and week precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53.

[18] A. A. Nasef and T. Noiri, Some weak forms of almost continuity, Acta Math. Hungar., 74 (3) (1997), 211-219.

[19] G. B. Navalagi, Pre-neighbourhoods, the Mathematics Education, 32 (4), Dec. (1998), 201-206.

[20] G. B. Navalagi, Further Properties of Pre-T0, Pre-T1,and Pre-T2 Spaces, In- ternational Journal of Mathematics and Computing Applications Vol. 3, Nos. 1-2, January-December 2011, pp. 67-75.

[21] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15 (3) (1965), 961-970.

[22] T. Noiri, Strongly θ-precontinuous functions, Acta Math. Hungar., 90 (4) (2001), 307-316.

[23] J. H. Park and Y. B. Park, On sp-regular spaces, J. Indian Acad Math., Vol. 17 (2) (1995), 213-218.

[24] M. Pal and P. Bhattacharyya, Feeble and strong forms of pre-irresolute func- tion,Bull. Malaysian Math. Soc. (Second Series), 19 (1996), 63-75.

[25] R. Paul and P. Bhettecharyya, On pre-Urysohn spaces, Bull. Malaysian Math. Soc., (Second Series) 22(1999) 23-34.

[26] I. L. Reilly and M. K. Vamanmurthy, On α-continuity in topological spaces, Acta. Math. Hungar., 45 (1-2) (1985), 27-32.

[27] H. A. Shareef, SP -open sets, SP -continuity and SP -compactness in topological spaces, M. Sc. Thesis, Sulaimani University 2007.

[28] M. H. Stone, Applications of the theory of boolean rings to topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.

[29] J. Tong, A separation axioms between T0 and T1, Ann. Soc. Sci. Bruxelles 96 (2) (1982), 85-90.

[30] N. V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl., 78 (2) (1968), 103-118.

[31] R. H. Yunis, Properties of θ-semi-open sets, Zanco J. of Pure and Applied

Sciences, 19 (1) (2007), 116-122.