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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