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.


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