The  purpose  of the  present work is to define and  study  new classes of the  most well-known concepts  of topological space, such as continuity  and  separation  ax- ioms called g*bp-continuity  and g*b-separation axioms by using g*b-set.

At first, some new types of separation  axioms, called g*b-Tk  and g* b-Dk   for k =

0, 1, 2 spaces, are defined in addition  to the space g*b-T 1 . Then,  relations  among these spaces are given in addition  to the investigation  of several of their properties and characterizations. Furthermore, the concept of g*b-R0  and g*b-R1  spaces are introduced  and studied.

After that, the notions of  g* bp-(almost g?bp-, weakly g*bp- and contra g*bp-) continuity are introduced.   We prove that  if a function f : X → Y  is g*b-continuous and Y is p-regular,  then  f is g*bp-continuous.   We also prove that  if f : X  → Y   is a g*bp-continuous injection and Y  is pre-T1, then  X is g*b-T1.

For a function f : X → Y , if f −1(C lA) is g*b-open set in X for each preopen set A in Y , then f is weakly g? bp-continuous.  For a function f : X → Y , we proved that weakly g*bp-continuous and almost g*bp-continuous are equivalent whenever Y  is almost p-regular space.  The functions with g?bp-closed graphs have been studied. Characterizations and properties  of these functions are obtained.

Finally, we define a multifunction F : X → Y  to be upper (lower) g*bp-continuous at  a point x ∈ X  if for each preopen set A in Y   containing  F (x)  ,there  exists a g*b-open  set U of X  containing  x such that  F (U ) ⊆ A (F (u) ∩ A = φ for ev- ery u ∈  U ).  We have given several properties  of upper  (lower) (almost,weakly) g*bp-continuous and contra  g*bp-continuous multifunction.


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