Republic of Iraq Ministry of Higher Education & Scientific Research AL-Qadisiyah University College of Computer Science and Department of Mathematics

Bornological Topology Space Separation Axioms A Research Submitted by Deyar

To the Council of the department of Mathematics ∕ College of Education, University of AL-Qadisiyah as a Partial Fulfilment of the Requirements for the Bachelor Degree in Mathematics

Supervised by

Fatma Kamel Majeed

A. D. 2019 A.H. 1440 Abstract

we study Bornological Topology Separation Axioms like bornological topology , bornological topology , bornological topology , bornological topology , bornological topology and the main propositions and theorems about this concept. introduction

For the first time in (1977), H. Hogbe–NIend [1] introduced the Concept of on a set and study Bornological Construction. In chapter one study Bornology on a set , Bornological subspace, convex , Bornological and . Bornological topology space were first introduced and investigated in [4], we introduce in chapter two Bornological topology space and we study Bornological topology continuous and bornological topology homeomorphism. Bornological topology open map, bornological topology separation axioms studied in chapter three like bornological topology , bornological topology , bornological topology And bornological topology Bornological topology and main properties have been studied.

The Contents

Subject Page

Chapter One

1.1 Bornological Space 1

1. 2 Bornivorous Set 4

Chapter Two

2.1 Bornological 6

2.2 Bornological Topology Continuous 8

Chapter three

3.1 Bornological topology And Bornological 9 topology

3.2 Bornological topology , Bornological topology 10 And Bornological topology

Chapter One

1.1 Bornological space In this section, we introduce some definitions, bornological space, bornological vector space, convex bornological vector space, separated bornological vector space, bounded map and some examples .

Definition 1.1.1[1]: Let A and B be two subsets of a vector space E. We say that:

i. A is circled if λA A whenever λ and   1. ii. A is convex if λ A+ μ A A whenever λ and μ are positive real numbers such that λ+μ=1. iii. A is disked, or a disk, if A is both convex and circled.

iv. A absorbs B if there exists αR, α>0, such that λA B whenever α  .

v. A is an absorbent in E if A absorbs every subset of E consisting of a single point.

Remark 1.1.2 [1]

(i) If A and B are convex and λ, μ K , then λA+μB is convex.

(ii) Every intersection of circled (respectively convex, disked) sets is circled (respectively convex, disked).

(iii)Let E and F be vector spaces and let u : E →F be a . Then the image, direct or inverse, under u of a circled (respectively convex , disked) subset is circled (respectively convex, disked).

Definition 1.1.3[3]: Let X be a topological space and let x . A subset N is said to be a neighborhood of x iff there exists open set G such that xG N. The collection of all neighborhoods of x is called the neighborhood system at x and shall be denoted by

N x .

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Remark 1.1.4[3]

Let X be then

(i) Every topological vector space X has a local base of absorbent and circled sets.

(ii) If A is circled in , x cl (A) and 0  |  | 1, then cl (A).

Remark 1.1.5[3]

A subset A is bounded iff it is absorbed by any neighborhood of the origin.

Definition 1.1.6[4]: A metrizable space is a topological space X with the property that there exists at least one metric on the set whose class of generated open sets is precisely the given topology.

Definition 1.1.7[1]: A bornology on a set X is a family ß of subsets of satisfying the following axioms:

(i) ß is a covering of , i.e. =  B ; B

(ii) ß is hereditary under inclusion i.e. if A ß and B is a subset of contained in A, then B  ß;

(iii)ß is stable under finite union.

A pair ( , ß) consisting of a set and a bornology ß on is called a bornological space, and the elements of ß are called the bounded subsets of .

Example 1.1.8 [1]: Let R be a with the absolute value. The collection:

ß = { is bounded subset of in the usual sense for the absolute value}. Then ß is a bornology on called the Canonical Bornology of .

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Definition 1.1.9[1]: Let be a vector space over the field and ß be a bornology on then ß is called on .If ß is stable under vector addition, homothetic transformations and the formation of circled hulls, in other words, if the sets ,

⋃| | belongs to ß whenever and belong to ß and . The pair ( , ß) is called a bornological vector space.

Definition 1.1.10[1]: A bornological vector space is called a convex bornological vector space if the disked hull of every is bounded i.e. it is stable under the formation of disked hull.

Definition 1.1.11[1]: A separated bornological vector space ( , ß) is one where { } is the only bounded vector subspace of .

Theorem 1.1.12[1] : A bornological vector space E is separated if and only if the vector subspace { } is b- closed in .

Example 1.1.13[1]: The Von–Neumann bornology of a topological vector space, Let E be a topological vector space. The collection ß ={A  E: A is a bounded subset of a topological vector space E}forms a vector bornology on E called the Von–Neumann bornology of E. Let us verify that ß is indeed a vector bornology on E, if ß 0 is a base of circled neighborhoods of zero in E, it is clear that a subset A of E is bounded if and only if for every B  ß there exists λ>0 such that A λ B. Since every neighborhood of zero is absorbent, ß is a covering of E. ß is obviously hereditary and we shall show that its also

/ / stable under vector addition. Let A 1 , A 2  ß and B 0 ß ; there exists B 0 such that B 0 + B B proposition (1.2.11). Since A and A are bounded in E, there exists positive scalars λ and μ such that A λ B and A μ B . with α= max(λ, μ), we have:

A +A λ B + μ B  α B + α B α(B + B ) α B .

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Finally, since ß 0 is stable under the formation of circled hulls (resp. under homothetic transformations). Then so is ß, and we conclude that ß is a vector bornology on E. If E is locally convex, then clearly ß is a convex bornology. Moreover, since every topological vector space has a base of closed neighborhoods of 0, the closure of each bounded subset of E is again bounded.

Definition 1.1.14[1]: Let X and Y be two bornological spaces and u: X → Y is a map of into Y. We say that u is a bounded map if the image under u of every bounded subset of is bounded in Y i.e. u(A) ß y ,  A ß x . Obviously the identity map of any bornological space is bounded.

1. 2 Bornivorous Set

In this section, we introduce the definition of bornivorous set and study the properties of it.

Definition 1.2.1: A sub set A of a bornonological vector space E is called bornivorous if it absorbs every bounded subset of E

Proposition 1.2.2: Let E be a topological vector space with Von Neumann bornology, then every neighbourhood of 0 is bornivorous set.

Proof:

Let A be neighbourhood of 0since E is a topological vector space then every bounded subset of E is absorbed by each neighbourhood of 0, i.e each neighbourghood of 0 absorbed every bound set of E.Then neighbourghood of 0 is bornivorous set.

Proposition 1.2.3: Let E be a metrizable topological vector space then a circled set that absorbs every sequence converging to 0 is a neighbourhood of 0 and every bornivorous subset of E is a neighbourhood of 0.

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Proof: Let A be a circled set absorbs every sequence converging topologically to 0.

Since E is a materizable topological vector space then every topologically convergent sequence is bornologically convergent.Then A is a circled set absorbs every sequence converging bornologically to 0. Since every neighbourhood of 0 absorbed every bounded set of E.By definition thus A is a neighbourhood of 0.

Let B be bornivorous subset of E by definition 1.2.1 B absorbs every bounded subset of E.E is a metrizable topological space then every topologically convergent sequence is bornologically convergent.Since each neighbourhood of 0 absorbed every bounded set of E. Then B is neighbourhood of 0.Then every bornivorous subset of E is a neighbourhood of 0.

Proposition 1.2.4: Let E be a bornological vector space then:

i. Every bornivorous set contains 0. ii. Every finite intersection of bornivorous set is bornivorous. iii. If B is bornivorous and then A is bornivorous .

Proof:

i. It is clear from proposition 1.2.2.

ii. Let Bi is bornivorous set for every i=1,…,n the set Bi absorbs every bounded subset

n n of E then absorbs every bounded subset of E then is Bi Bi i1 i1 bornivorous set. iii. Let B is bornivorous set then B absorbs every bounded subset of E since then absorbs every bounded subset of E, thus A is bornivorous set.

Proposition 1.2.5: Let E,F be bornological vector space and let be a bounded linear mab then the inverse image of a bornivorous subsets of F is bornivorous in E.

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Chapter Two

2.1 Bornological Topology space:- In this section we define bornological open set depended on bornivorous set and construct bornological topological space and some important definitions.

Definition 2.1.1: Let E be a bornonological vector space , subset A of E is called bornological open ( for brief b-open) set, if the set { } is bornivorous for every A, .The complement of bornological open set is called bornological closed (for brief b-closed) set.

Proposition 2.1.2:

i. The finite intersection of b-open sets is b-open set. ii. The union of b-open sets is b-open set.

Proof:

(i)Let A1,A2,…,An be b-open sets then by definition 2.1.1 the set

{ } (such that j is counter of elements of Ai ) is

bornivorous set and then { } { } { }

{ }.Since { } is bornivorous set then the finite

intersection of bornivorous set is bornivorous set and then { }

is bornivorous .Thus b-open set.

(ii).Let Aλ be b-open set such that then

{ } since { } ⋃

{ } ⋃ { } is bornivorous set.

Thus ⋃ is bornivorous set.

Remark 2.1.3: As a consequence of the proposition 2.1.2 the family of all bornological open(b-open) sets define a topology on E called the

6 bornological topology denoted by b- topology and the pair (E,T) called bornological topological space( for brief b-topological space).

Definition 2.1.4: Let (E,T) be a bornological topological space a collection U of subsets of E is said to form a bornological base denoted by(b- base) for b-topology T if ,

1. . 2. For each point and each b-open set A of there exist some such that .

Definition 2.1.5 : Let (E,T) be a bornological topological space a collection of subsets of E is called bornological sub base denoted by

(b-subbase) for the b-topology T if and finite intersection of member of form a b-base of b-topology T .

Definition 2.1.6 : Let E be a bornological topological space and the intersection of all b-closed subsets of E containing A is called bornological closure set (b-closure set) denoted by ̅ .

Definition 2.1.7: Let E be a bornological topological space and . A point is said to be bornological interior point of A if there exists b-open set A1 such that . The set of all bornological interior points of A is called bornological interior of A and denoted by b-Int(A).

Definition 2.1.8 :

Let (E,b- T)be a bornological topology space and such that Y is bornivorous set. A bornological topology space of Y is the collection b- closure TY given by b-closure { } the bornological space (Y , b- TY) is called bornological topology sub space of (E,b- T).

7 2.2 Bornological Topology Continuous:-

In this section we introduce the concept of bornological topology continuous in bornological topology space and Bornological Topology Homeomorphism

Definition 2.2.1: Let and be b-topology spaces. A function is said to be bornological topology continuous (b- topology continuous) at iff every b-open in containing there exists b-open set in containing such that . is said to be b-topology continuous iff it is b-topology continuous for each point of .

Definition 2.2.2: Let and be b-topology spaces,and let be a function of into . Then

1. is said to be a b-topology open if is b-open in when ever is b-open in 2. is said to be b- topology closed iff is b-closed in when ever is b-closed in 3. is said to be b- topology homeomorphism iff, i. is one-one and onto. ii. is b- topology continuous. iii. is b- topology continuous.

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3.1 Bornological topology And Bornological topology In this section, we study Bornological Topology Separation

Axioms like bornological topology , bornological topology and the main propositions and theorems about this concept.

Definition 3.1.1: A b-topology space is said to be a bornological topology space if and only if given any distinct points there exist a b-open set such that and or there exists a b- open set such that and .

Example 3.1.2: Let and be two b-topology spaces on and let be finer than . If is a space then is space.

Proposition 3.1.3: Every sub space of a space is a space.

Proof: Let be a space and let be a sub space of

. Let be two distinct points of . Since are also distinct points of . Since is a space , there exists a b- open A in of and . Then is a b-open set in such that and . Hence is a space.

Proposition 3.1.4: A b-topology space is a space if and only if for any distinct points , the b – closure of and are distinct.

Proof : Let ̅̅ ̅̅ ̅̅ ̅̅ where are points of . Since ̅ ̅ ̅̅ ̅̅ ̅̅ , there exists at least one point which belongs to one of them, say ̅̅ ̅̅ , and does not belong to ̅̅ ̅̅ . We claim that ̅̅ ̅̅ .

Let ̅̅ ̅̅ . Then ̅̅ ̅̅ {̅ ̅̅ ̅̅̅ ̅̅̅ ̅̅̅̅ ̅ } ̅̅ ̅̅ , and so ̅̅ ̅̅ ̅̅ ̅̅ which is a contradication.

Accordingly ̅̅ ̅̅ and consequently ̅̅ ̅̅ . Also since ̅̅ ̅̅ is b-closed, ̅̅ ̅̅ is b-open set. Hence ̅̅ ̅̅ is a b-open set containing and not containing , then is

space. 9 Conversely, Let be a space and let be two distinct points of . Then we have to show that ̅̅ ̅̅ ̅̅ ̅̅ . Since the space is b , there exists an b-open set containing one of them, suppose , but not containing . Then is a b-closed set which does not contain but contains . By (definition 1.6) , ̅̅ ̅̅ is the intersection of all b-closed sets containing . It follows that ̅̅ ̅̅ . Hence implies that ̅̅ ̅̅ . Thus ̅̅ ̅̅ but ̅̅ ̅̅ . It follows that ̅̅ ̅̅ ̅̅ ̅̅ . Thus it is shown that in a space distinct points have distinct closures.

Definition 3.1.5: A b-topology space is said to be a space if and only if given any pair of distinct points and of there exists two b-open sets and such that : but and but .

Proposition 3.1.6: Every b-subspace of a space is a space.

Proof:- Let be a space and let ( ) be a b-subspace of

. Let be two distinct points of . Since are also distinct points of. . Since is a space , there exist b- open sets and such that but and but . Then and are b-open sets such that : but

and but . We have ( ) is a space.

Example 3.1.7: Every bornological topology space b-finer than a on any b-topology space is a space.

3.2 Bornological topology , Bornological topology And Bornological topology

Definition 3.2.1: A b-topology space is said to a be a space iff for every pair of distinct points of there exists b-open of and of such that .

Example 3.2.2: Let is a space, if be a b-topology space on and b-finer than then is also space.

Solution: Let be any two distinct points of since is a space , there exists b-open sets in , such that and . Since is b-finer than , are also b-open sets in such that and . Hence is also space. 10 Proposition 3.2.3: Every b-subspace of space is space.

Proof:- Let be a space and let be a non-empty subset of . Let be two distinct points of then are also distinct points of . Since is space there exists disjoint b-open sets in of and respectively. By (definition 1.8) and are b-open sets in . Also and then and and then and since , we have

Thus and are disjoint b-open set in of and respectively. Hence is space.

Proposition 3.2.4: Every space is a space.

Proof:- Let be a space and let be any two distinct points of . Since the space is , there exists a b-open set of and a b-open of such that . Hence the space is space.

Proposition 3.2.5: Let be a space. Let be a one to one, onto and b-open function from to then is space.

Proof:- Let be two distinct points of . Since is one-one on to map, there exists distinct point of such that and .

Since is space, there exist b-open sets in and such that:

and . Now since is a b-open mapping, and are b-open sets in such that:

and (since is one-one). Then

is space. Proposition 3.2.6: Let be b-topology space and be a

space. Let be a one-one b- then

is a space. 11 Proof:- Let be any two distinct points of . Since is one-one , . Let so that

and . Then such that .

Since is a space there exists b-open sets in , and such that and and . Since b-continuous . and are b-open sets in .

Now and

.

Thus for every pair of distinct points of there exist disjoint b- open sets if and such that and

we have is space. Proposition 3.2.7: Let be a b-topology space and let be a

space. If are b-continuous function of into then is a b-closed subset of .

Proof:-

Let then .By (1) we have .

Thus are two distinct points of . Since is a space there exist b-open sets such that and .

, so that

.

Since are -continuous functions, and are - open sets in and is a - open set in containing . Let

. Then and . We have and .………...……………………………………….(2)

Since it follows from (2) that and so by (1),

. Thus we have shown that . Therefore . Hence - contains a - open set of each of its points and so is b-open set in . It follows that is a - closed subset of .

12 Corollary 3.2.8: Let be a - space and let f be a -continuous function of into itself. Then is - closed set of .

Proof:- from (proposition 4.13) with and replaced by the identity function.

Definition 3.2.9: A -topology space is said to be - space if and only if for every -closed set in and every point , there exists - open sets and in such that , and .

Proposition 3.2.10: A -topology space is - space if for every point and every - open set containing there exists a - open set containing such that ̅ .

Proof:- Let be any - open set of . Then there exists - open set such that . Since is - closed and . By (definition

4.16) there exists - open sets , such that , and

so that ̅ ̅̅ ̅ ……………………….…………(1)

Also …………………….……….. (2) From (1) and (2), we get ̅ .

Conversely, Let be any - closed set and let . Then . Since is a - open set containing by given there exists a - open set such that and ̅ ̅ . Then ̅ is a -open set containing . Also ̅ . Hence is -

space

Proposition 3.2.11: Every - space is a - space.

Proof:- Let be a - space and be two distinct points of . Since then every - open set containing and there exists a - open set containing such that ̅ (proposition (4-14)).

Thus ̅ and since ̅ closed set containing not , then there exists - open set , and ( is - space ). Hence is - space.

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Proposition 3.2.12: Let be a - space if be a - homomorphism then be a - space. Proof:- Let be a -closed subset of and let be a point of such that . Since is a one- one, on to function, there exists such that . Since is - continuous function is - closed set in . Also . Thus is a - closed set in and is a point of such that

. Since is - space, there exists - open sets and

such that , and .

, and (since is one- one). Also since is a - open function, and are - open sets in . Thus there exists - open sets and

such that and . Then is also - space.

Proposition 3.2.13: Let be a - space and be a - subspace of then is -111 space. Proof:- Let be a -closed subset of and be a point of such that

. Then ̅ ̅ . Since is -closed set in , we have ̅ so that ̅ ………………………….(1)

Therefore ̅ ̅ . Thus ̅ is a -closed subset of such that ̅ . Since is a - space, there exists - open set and such that , ̅ and . Since , ̅ ̅ by (1). Also . Thus is a - space

Definition 3.2.14: A -topology space is said to be - if and only if for every pair , of disjoint -closed sets of there exists - open sets , such that , and .

14 Proposition 3.2.15: A b- topology space is - if and only if for any -closed and -open set containing , there exists a -open set such that and - ̅ .

Proof:- Let be a - space and let be any - closed set and a - open set such that . Then is a -closed set such that

. Thus and are disjoint - closed subset of . Since the space is -

, there exists -open sets and such that , and

so that . ̅ ̅̅ ̅ But - - ( is closed ) …………………(1)

Also …………………………………………….(2) from (1)and (2), we get - ̅ . Thus there exists a - open set such that and - ̅ .

Conversely, Let and be -closed subsets of such that

so that . Thus the -closed set is contained in the -open set . Then there exists a -open set such that and -

̅ which implies - ̅ . Also - ̅ . Thus

and - ̅ are two disjoint open set such that and ̅ Then the space is - .

Proposition 3.2.16: Let be a - space and -homomorphism then is a - space.

Proof:- Let be a disjoint -closed subset of . Since is a - continuous function and are -closed subset of . Also

.Thus

is a disjoint of -closed subsets of . Since the space is - space there exists -open sets and of such that

and . But

and we have . Also since is a -open function, and are b-open subsets of , such that (since is one-one). Thus there exists -open subsets and of such that and . Then is also a - space.

15 Proposition 3.2.17: Every - space is also a - space.

Proposition 3.2.18: Let be a - space and any -closed b-subspace of then is - space.

Proof:- Let be disjoint -closed subset of . Then there exists - closed subsets , of such that and .

Since is -closed it follows that are disjoint -closed subsets of . Then there exists -open subsets of such that

and . ( be a - space ). Since and , these relations imply that

and . Then

are -open subsets of . Such that and

. We have is a - space.

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References

[1] H. Hogbe Nlend, Bornologies and , Netherlands: North –Holland Publishing Company Netherlands, 1977.

[2] S. a. D. P. Dierolf, "Bornological Space Of Null Consequences," Arch. Math., vol. 65, no. 1, pp. 46-52, 1995.

[3] O. S. V.I. Bogachev, Topological Vector Spaces and Their Applications, Springer Nature, 2017.

[4] F. K. M. Al-Basri, "On Bornivorous Set," Journal of AL-Qadisiyah for pure science, vol. 11, no. 1, pp. 139-147, 2014.

[5] A. N. D. Al-Salhi, "On Bornological Constructions," College of education journal, no. 1, pp. 131-141, 2007.

[6] F. K. M. Al-Basri, "The Relationship Between Boronological Convergence of Net and Topological Convergence of Net," Journal of AL-Qadisiyah for computer science and mathematics, vol. 6, no. 2, pp. 65-76, 2014.

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