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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