**Abstract:**

The purpose of the present work is to introduce and investigate a new class of sets called Pγ -open sets and use this class to define and study new classes of the most well-known concepts topological space such as continuity and new types of separation axioms.

At the beginning of this work, we define the class of Pγ -open sets which is contained in the class of preopen sets and contains the class of γ-open sets. It is shown that the family of Pγ -open sets form a supratopology on a space X , we prove that the family of Pγ -open sets form a topology on a space X if γ is a pre regular operation. We prove that the family of Pγ -open sets and preopen sets are identical if γ is a pre identity operation, and also in pre γ-regular spaces, while the family of γ-open sets and Pγ -open sets coincide in submaximal spaces. The study shows that if X is γ-regular, then τ ⊆ Pγ O(X ) and for any subset A of a space X we have pγ Bd[pγ Bd{pγ Bd(A)}] = pγ Bd[pγ Bd(A)]. In the last section, Pγ -g.closed sets are defined and giving characterization theorems to some spaces.

Further more, the notions of Pγ -continuous and weakly Pγ -continuous are intro- duced. It is shown that Pγ -continuity is weaker than γ-continuity in the sense of Basu et al., as well as being stronger than precontinuity. It is noticed that weakly Pγ -continuity is weaker than Pγ -continuity and stronger than weakly precontinu- ity. In addition, several conditions are given which make Pγ -continuity and weakly Pγ -continuity equivalent to some other types of continuity. The functions with Pγ – closed graphs have been studied to obtain some characterizations and properties of it.

Finally, some new types of separation axioms are defined, we show that Pγ -T1 gives Pγ -T 1 2

and Pγ -T 1 gives Pγ -T0. It is shown that a space X is Pγ -T 1 2 if and only if every singleton is either Pγ -closed or Pγ -open. We prove that the notions of Pγ -D1 and Pγ -D2 are equivalent. Further more, the properties and characterizations of Pγ -R0 and Pγ -R1 spaces have been studied.

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