Some Type of Separation Axioms in Bornological Topological Space

In this paper, we introduce the concept of bornological topological continuous in bornological topological space and study some its properties. Also we define bornological topological open function, bornological topological closed function and bornological topological homeomorphism function and investigate some new properties of them. Finally some separation axioms have been studied in bornological topological space like B − τ0, B − τ1, B − τ2, B − τ3, B − τ4 and the relationships among them.


Introduction
Modern analysis is developed by the setting of functional analysis is done on various topological structres, all these spaces are spacial cases of uniform spaces, one of these spaces by use bornological structre to define a topological spaces. In 1977, Hogbe-Nlend [7] introduced the concepts of bornology on a set. Noiri. [8] in 1984 foundα-continuous functions. In 1999 [7] Balasubramanian, used fuzzy set to detrmined fuzzy β-open sets and fuzzy β-separation axioms. In 1995, Dierolf and Domanski [6] studied various bornological properties and Bornological Space Of Null Sequences. Barreira and Almeida [5] in 2002 introduced Hausdorff Dimension in Convex Bornological Space. Since that time, several methods for constructing new bornologies like forming products, subspace and quotient bornologies like that which were presented in 2007 [2] by Al-Deen and Al-Shaibani, The space of entire functions over the complex field C was introduced by Patwardhan who defined a metric on this space by introducing a real-valued map on. In 2018 Al-Basri [8] found the relationship between the sequentially bornological continuous map in bornological vector spaces (bvs) and sequentially bornological compact spaces have been investigated and studied as well as between them and bornological complete space in 2018 [8]. In this paper we study bornological topology spaces, intrduce B-open set and B -closed set and some concepts have been defined depending on bornivorous set. The bornivorous set is the subset N of a bornonological vector space E if it absorbs every bounded subset of E. So that many researchers investigated new properties like,B -base, B -sub base, B -closures set,B -interior set, B -subspace. this paper contains 6 sections, introducation of this paper in section 1, section 2 contains some basic concepts of bornological space, section 3 we introduce the concept of bornolological topology continuous (written B-topology continuous), bornological topology space and bornological topology open (written B-topology open), bornological topology closed (written B-topology closed). section 3 we define bornological topology homeomorphism (written B -topology homeomorphism), some new properties of, bornological topology open, bornological topology closed and Conversely, let the condition hold and let N be

Bornolopical Topological Homeomorphism
We define bornological topology open function, bornological topology closed function in this section and we define bornological topology homeomorphism function and investigate some properties on them.
(2) f is said to be B-topology closed iff and denoted by ( Proof. Since f is one-to-one and f N is also one-to-one and

Bornological Topological Separaeion Axioms
, there exists at least one point z of E which belongs to one of them, say B − (x), and does not belong to B − (y). We claim that Proof. Let x, y be any two distinct points of E since (E, τ 0 ) is a B − τ 2 space, there exists B -open sets in τ 0 , A 0 , A 1 such that x ∈ A 0 , y ∈ A 1 and A 0 ∩ A 1 = φ. Since τ 1 is B-finer than τ 0 . Then A 0 , A 1 are also B-open sets in τ 1 such that x ∈ A 0 , y ∈ A 1 and A 0 ∩ A 1 = φ. Hence (E, τ 1 ) is also B − τ 2 space. .
Proof. Let y 1 , y 2 be two distinct points of F . f is one-to-one, on-to map, there exists distinct point f (N ) and f (M ) are B -open sets in F such that: Proof. Let x 1 , x 2 be any two distinct points of E.

Conclusion
In this paper we find new structure by use bornological space, we introduce the structre of brnological topological space. we define bornological topological open function, bornological topological closed function and bornological topological homeomorphism function, bornological topological continuous and investigate some new properties of them. Separation axioms have been studied in bornological topological space like B − τ 0 , B − τ 1 , B − τ 2 , B − τ 3 , B − τ 4 and the relationships among them, also we study topological properties and heredity property under exist this axioms.