On Almost β Topological Vector Spaces

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ISSN Online: 2333-9721 ISSN Print: 2333-9705

DOI: 10.4236/oalib.1105408 May 29, 2019 1 Open Access Library Journal

On Almost

β

-

Topological Vector Spaces

Shallu Sharma, Sahil Billawria, Madhu Ram, Tsering Landol

Department of Mathematics, University of Jammu, Jammu and Kashmir, India

Abstract

In this paper, we have introduced a new generalized form of topological vec-tor spaces, namely, almost β-topological vector spaces by using the concept of

β-open sets. We have also presented some examples and counterexamples of almost β-topological vector spaces and determined its relationship with to-pological vector spaces. Some properties of β-topological vector spaces are also characterized.

Subject Areas

Mathematical Analysis

Keywords

β-Open Sets, δ-Open Sets, Regular-Open Sets, Almost β-Topological Vector Spaces

1. Introduction

The concept of topological vector spaces was introduced by Kolmogroff [1] in 1934. Its properties were further studied by different mathematicians. Due to its large number of exciting properties, it has been used in different advanced branches of mathematics like fixed point theory, operator theory, differential calculus etc. In 1963, N. Levine introduced the notion of semi-open sets and semi-continuity [2]. Nowadays there are several other weaker and stronger forms of open sets and continuities like pre-open sets [3], precontinuous and weak precontinuous mappings [3], β-open sets and β-continuous mappings [4],

δ-open sets [5], etc. These weaker and stronger forms of open sets and continui-ties are used for extending the concept of topological vector spaces to several new notions like s-topological vector spaces [6] by M. Khan et al. in 2015, irre-solute topological vector spaces [7] by M. Khan and M. Iqbal in 2016,

β-topological vector spaces [8] by S. Sharma and M. Ram in 2018,

al-How to cite this paper: Sharma, S., Billawria, S., Ram, M. and Landol, T. (2019) On Almost β-Topological Vector Spaces Open Access Library Journal, 6: e5408.

https://doi.org/10.4236/oalib.1105408

Received: April 16, 2019 Accepted: May 26, 2019 Published: May 29, 2019

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/oalib.1105408 2 Open Access Library Journal

mosts-topological vector spaces [9] by M. Ram et al. in 2018, etc. The aim of this paper is to introduce the class of almost β-topological vector spaces and present some examples of it. Further, some general properties of almost β-topological vector spaces are also investigated.

2. Preliminaries

Throughout this paper,

(

X,τ

)

(or simply X) and

(

Y,σ

)

(or simply Y) mean topological spaces. For a subset A X⊆ , Cl A

( )

denotes the closure of A and

( )

Int A denote the interior of A. The notation  denotes the field of real numbers  or complex numbers  with usual topology and

ε

, η

represent the negligibly small positive numbers.

Definition 2.1 AsubsetAofatopologicalspaceXissaidtobe: 1) regular open if A Int Cl A=

(

( )

)

_.

2) β-open [4]if A Cl Int Cl A⊆

(

( )

)

.

Definition 2.2 AsubsetAofatopologicalspaceXissaidto beδ-open [5]if foreach x A∈ _,_there_exists_a_regular_open_set_U_in_X_such_that x U∈ ⊆ A.

The union of all β-open (resp. δ-open) sets in X that are contained in A X⊆

is called β-interior [10] (resp. δ-interior) of A and is denoted by βInt A

( )

(resp.

( )

Int Aδ ). A point x is called a β-interior point of A X⊆ if there exists a

β-open V in X such that x V∈ ⊆A_{. The set of all}_β_{-interior points of}_A_{is equal}

to βInt A

( )

. It is well known fact that a subset A X⊆ is β-open (resp. δ-open) if and only if A=βInt A

( )

(resp. A Int A= δ

( )

). The complement of β-open

(resp. δ-open, regular open) set is called β-closed (resp. δ-closed [5], regular closed). The intersection of all β-closed (resp. δ-closed) sets in X containing a subset A X⊆ _{is called}_β_-closure_[10]_(resp._δ_{-closure) of}_A_{and is denoted by}

( )

Cl A

β (resp. Cl Aδ

( )

). It is also known that a subset A of X is β-closed (resp.

δ-closed) if and only if A=βCl A

( )

(resp. A Cl A= δ

( )

). A point x∈βCl A

( )

if and only if A V_ ≠φ_{for each}_β_{-open set}_V_in_X_containing_x_{. A point}

( )

x Cl A∈ δ if A Int Cl O

(

( )

)

≠φ for each open set O in X containing x.

The family of all β-open (resp. β-closed, regular open) sets in X is denoted by

( )

O X

β (resp. βC X

( )

, RO X

( )

). If A∈βO X

( )

, B∈βO Y

( )

, then

(

)

A B× ∈βO X Y× (with respect to the product topology). The family of all

β-open sets in X containing x is denoted by βO X x

(

,

)

.

Definition 2.3 [11]Afunction f X: →Y _from _a_topological _space _X_to _a

topologicalspaceYiscalledalmostβ-continuousat x X∈ _if_for_each_open_set

O of Y containing f x

( )

, there exists V∈βO X x

(

,

)

such that

( )

(

( )

)

f V ⊂Int Cl O _.

Also we recall some definitions that will be used later.

Definition 2.4 [12]LetTbeavectorspaceoverthefield . Let

τ

_be_a to-pologyonTsuchthat

1) For each x y T, ∈ _{and each open neighborhood O of}x y+ _{in T}, there exist open neighborhoods O1 and O2 of x and y respectively in T such that

1 2

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2) For each λ ∈_, x T∈ _{and each open neighborhood O of}λx_{in T},

there exists open neighborhoods O1 of λ in  and O2 of x in T such that 1 2

O O ⊆O_.

Then the pair

(

T_{( )}_,τ

)

is called topological vector space.

Definition 2.5 [8]Let Tbeavectorspace overthefield . Let

τ

1) For each x y T, ∈ and each open neighborhood O of x y+ in T, there exist β-open sets V1 and V2 in T containing x and y respectively such that

1 2

V V+ ⊆O_,_and

2) For each λ ∈__, x T∈ _{and each open neighborhood O of}λx_{in T}_,

there exist β-open sets V1 containing λ in  and V2 containing x in T

such that V V1⋅ 2⊆O.

Then the pair

(

T( ),τ

)

is called β-topological vector space.

Definition 2.6 [13]LetTbeavectorspaceoverthefield . Let

τ

1) For each x y T, ∈ and each regular open set U T⊆ _containing x y+ _,

there exist pre-open sets P1 and P2 in T containing x and y respectively such

that P P1+ 2 ⊆U , and

2) For each λ ∈__, x T∈ _{and each regular open set}U T⊆ _containing

x

λ _,_{there exist pre-open sets}P₁ in  containing λ_and P₂ containing x in T such that P P1⋅ 2 ⊆U .

Then the pair

(

T( ),τ

)

is called an almost pretopological vector space.

Definition 2.7 [9]Let Tbeavectorspace overthefield . Let

τ

1) For each x y T, ∈ and each regular open set U T⊆ _containing x y+ _,

there exist semi-open sets S1 and S2 in T containing x and y respectively such

that S S1+ 2 ⊆U, and

2) For each λ ∈__, x T∈ _{and each regular open set}U T⊆ _containing

x

λ , there exist semi-open sets S1 in  containing λ and S2 containing x

in T such that S S1⋅ 2⊆U.

Then the pair

(

T( ),τ

)

is called an almost s-topological vector space.

3. Almost

β

-Topological Vector Spaces

In this section, we define β-topological vector spaces and present some examples of it.

Definition 3.1 Let Z be a vector space over the field  ( or  with standardtopology). Let

τ

_be_a_topology_on_Z_such_that

1) For each x y Z, ∈ _{and each regular open set}U⊆Z_containing x y+ ,

there exist β-open sets V1 and V2 in Z containing x and y respectively such

that V V1+ 2 ⊆U , and

2) For each λ ∈__, x Z∈ _{and each regular open set}U⊆Z_containing

x

λ , there exist β-open sets V1 in  containing

λ

and V2 containing x in

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Then the pair

(

Z( ),τ

)

is called an almost β-topological vector space.

Some examples of almost β-topological vector space are given below:

Example 3.1 Let Z= be the real vector space over the field , where

=_

 with the standard topology and

τ

_{be the usual topology endowed on}_Z_,

that is,

τ

_{is generated by the base} =

{

( )

a b a b, : , ∈_

}

. Then

(

Z( ),τ

)

is an

almost β-topological vector space. For proving this, we have to verify the fol-lowing two conditions:

1) Let x y Z, ∈ . Consider any regular open set U =

(

x y+ −,x y+ +

)

in Z

containing x y+ _{. Then we can opt for}_β_{-open sets}V₁=

(

x−η,x+η

)

and

(

)

2 ,

V = y−η y+η in Z containing x and y respectively, such that V V1+ 2⊆U

for each

2

η <_{. Thus first condition of the definition of almost}_β_-topological

vector space is satisfied.

2) Let λ ∈ =  and x Z∈ _{. Consider a regular open set}

(

,

)

U = λx−λx+ in Z =__containing λx_{. Then we have the following}

cases:

Case (I). If λ >0_and x>0_{, then}λ >x 0_{. We can choose}_β_{-open sets}

(

)

1 ,

V = λ η λ η− + in  containing λ_and V₂ =

₍

y−η,y+η

₎

in Z contain-ing x, such that V V1⋅ 2⊆U for each η<_λ_{+ +}_x ₁.

Case (II). If λ <0_and x<0_{, then}λ >x 0_{. We can choose}_β_{-open sets}

(

)

1 ,

V = λ η λ η− + in  containing λ_and V2=

(

x−η,x+η

)

in Z

contain-ing x, such that V V1⋅ 2⊆U for each η<₁_{− −}_λ _x.

Case (III). If λ >0_and x<0_(resp. λ <0_and x>0_{), then} λ <x 0_{. We}

can choose β-open sets V1=

(

λ η λ η− , +

)

in  containing λ and

(

)

2 ,

V = x−η x+η in Z containing x, such that V V1⋅ 2⊆U for each

1 x

η λ <

+ −

 _(resp.

1 x

η λ <

− +

 _).

Case (IV). If λ =0_and x>0_(resp. λ >0_and x=0_{), then} λ =x 0_{. We}

can select β-open neighborhoods V1= −

(

η η,

)

(resp. V1=

(

λ η λ η− , +

)

) in 

containing λ_and V2=

(

x−η,x+η

)

(resp. V2 = −

(

η η,

)

in Z containing x,

such that V V1⋅ 2⊆U for each η <_x₊₁ (resp. η< _λ₊₁).

Case (V). If λ =0_and x<0_(resp. λ <0_and x=0_{), then}λ =x 0_{. We}

can select β-open neighborhoods V1= −

(

η η,

)

(resp. V1=

(

λ η λ η− , +

)

) in 

containing λ_and V2=

(

x−η,x+η

)

(resp. V2 = −

(

η η,

)

in Z containing x,

such that V V1⋅ 2⊆U for each η <₁₋_x (resp. η<₁₋_λ).

Case (VI). If λ =0_and x=0_{, then}λ =x 0. Then for β-open neighbor-hoods V1= −

(

η η,

)

of λ in  and V2 = −

(

η η,

)

of x in Z, we have

1 2

V V⋅ ⊆U_{for each}η< .

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Example 3.2 Let Z = be the real vector space over the field  with the topology

τ

_{generated by the base}

( )

{

_{a b a b}, : ,

}

{

( )

_{c d}, c: ,_{c d}

}

= ∈   ∈

 , where _c_{denotes the set of}

ir-rational numbers. Then

(

Z_{( )}_,τ

)

is an almost β-topological vector space. Example 3.3 Consider the field _=__{with standard topology. Let}Z =_

be the real vector space over the field  endowed with topology

{ }

{

, 0 ,

}

τ = φ _ _{. Then}

(

Z_{( )}_ ,τ

)

is an almost β-topological vector space. Example 3.4 Let

τ

_{be the topology induced by open intervals}

(

a b,

)

and the sets

[

c d,

)

where a b c d, , , ∈__with 0< <c d _{. Let}Z =__{be the real}

vector space over the field  endowed with topology

τ

_{, where} _=__with

the standard topology. Then

(

Z( ),τ

)

is an almost β-topological vector space.

The above four examples are examples of almost β-topological vector spaces, we now present an example which don’t lie in the class of almost β-topological vector spaces.

Example 3.5 Let

τ

be the topology generated by the base

[

)

{

a b a b, : ,

}

= ∈_

 and let this topology

τ

_{is imposed on the real vector}

space Z =__{over the topological field}_=__{with standard topology. Then}

( )

(

Z_,τ

)

fails to be an almost β-topological vector space. For, U =

[

0,1

)

is reg-ular open set in Z containing 0= −1.0₍− ∈ =1 _ __and 0∈Z_{) but there do}

not exist β-open sets V1 in  containing −1 and V2 in Z containing 0 such

that V V1⋅ 2 ⊆U .

Remark 3.1 By definitions, it is clear that, every topological vector space is an almost β-topological vector space. But converse need not be true in general. For, examples 3.2 and 3.3 are almost β-topological vector spaces which fails to be to-pological vector spaces.

Remark 3.2 The class of almost pretopological vector spaces and almost s-topological vector spaces lie completely inside the class of almost β-topological vector spaces.

4. Characterizations

Throughout this section, an almost β-topological vector space

(

Z( ),τ

)

over

the topological field  will be simply written by Z and by a scalar, we mean an element from the topological field .

Theorem 4.1 LetAbeanyδ-opensetinanalmostβ-topologicalvectorspace Z. Then x A A+ ,λ ∈βO Z

( )

, foreach x Z∈ _and_each_non-zero_scalar λ_.

Proof. Let y x A∈ + . Then y x a= + _{for some}a A∈ _{. Since}_A_is_δ_-open,

there exists a regular open set U in Z such that a U∈ ⊆A_. ⇒ − + ∈x y U_.

Since Z is an almost β-topological vector space, there exist β-open sets V1

and V2 in Z such that − ∈x V y V1, ∈ 2 such that V V1+ 2⊆U. Now 2

x y x V U A

− + ∈ − + ⊆ ⊆ ⇒V₂⊆ +x A_{. Since} V₂ is β-open,

(

)

y∈βInt x A+ . This shows that x A+ =βInt x A

(

+

)

. Hence x A+ ∈βO Z

( )

. Further, let x∈λA_{be arbitrary. Since}_A_is_δ_{-open, there exists a regular}

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vec-DOI: 10.4236/oalib.1105408 6 Open Access Library Journal

tor space, there exist β-open sets V1 in the topological field  containing λ−1

and V2 in Z containing x such that VV1 2 ⊆U . Now λ−1x∈λ−1V2⊆A 2

V λA

⇒ ⊆ ⇒ ∈x βInt A

( )

λ and hence λA=βInt A

( )

λ . Thus λA _is

β-open in Z; i.e., λA∈βO Z

( )

.

Theorem 4.2 LetBbeanyδ-closedsetinanalmostβ-topologicalvectorspace Z. Then x B B+ ,λ ∈βCl Z

( )

_for_each x Z∈ _and_each_non-zero_scalar λ.

Proof. We need to show that x B+ =βCl x B

(

+

)

. For, let y∈βCl x B

(

+

)

be arbitrary and let W be any δ-open set in Z containing − +x y_{. By definition}

of δ-open sets, there is a regular open set U in Z such that − + ∈ ⊆x y U W_.

Then there exist β-open sets V1 and V2 in Z such that − ∈x V y V1, ∈ 2 and

1 2

V V+ ⊆U_{. Since}y∈βCl x B

₍

+

₎

, then by definition,

(

x B+

)

V2≠φ ⇒

there is some a∈

(

x B+

)

V2

(

2

)

(

1 2

)

x a B x V B V V B U B W

⇒ − + ∈  − + ⊆  + ⊆ ∩ ⊆  ⇒B W ≠φ. Thus

( )

x y Cl Bδ

− + ∈ . Since B is δ-closed set, we have, − + ∈ ⇒ ∈ +x y B y x B. Therefore x B+ =βCl x B

(

+

)

. Hence x B+ ∈βCl Z

( )

.

Next, we have to prove that λB=βCl B

( )

λ . For, let x∈βCl B

( )

λ be arbi-trary and let W be any δ-open set in Z containing _λ−1_x_{. By definition, there is a}

regular open set U in Z such that _λ−1_{x U W}_{∈ ⊆} _{. Then there exist}_β_{-open sets} 1

V containing _λ−1_{in topological field} __and 2

V containing x in Z such that

1 2

V V⋅ ⊆U_{. Since} x∈βCl B

( )

λ , then there is some a∈

( )

λB V2. Now

(

)

(

)

1 1

2 1 2

a B V B V V B U B W

λ− _∈ _ λ− _⋅ _⊆ _ _⋅ _⊆ _ _⊆ _ _⇒_{B W}_ _≠_φ_{. Thus}

( )

1_{x Cl B} _B _x _B

δ

λ− _∈ _{= ⇒ ∈}λ _{. Therefore} _λ_B₌_β_{Cl B}

_{( )}

_λ _{. Hence}

( )

B Cl Z

λ ∈β .

Theorem 4.3 ForanysubsetAofanalmostβ-topologicalvectorspaceZ, the followingassertionshold:

1) x+βCl A

( )

⊆Cl x Aδ

(

+

)

for each x Z∈ .

2) λβCl A

( )

⊆Clδ

( )

λA for each non zero scalar λ.

Proof. 1) Let z x∈ +βCl A

( )

. Then z x y= + for some y∈βCl A

( )

. Let O

be an open set in Z containing z, then z O Int Cl O∈ ⊆

(

( )

)

_{. Since}_Z_{is an almost}

β-topological vector space, then there exist V V1, 2∈βO Z

( )

containing x and y

respectively such that V V1+ 2 ⊆Int Cl O

(

( )

)

. Since y∈βCl A

( )

, then there is

some a A V∈  2. As a result, x a+ ∈

(

x A+

) (

 V V1+ 2

) (

⊆ x A+

)

Int Cl O

(

( )

)

(

x A

)

Int Cl O

(

( )

)

φ

⇒ +  ≠ . Thus z Cl x A∈ δ

(

+

)

. Therefore

( )

(

)

x+βCl A ⊆Cl x Aδ + .

2) Let x∈βCl A

( )

and let W be an open set in Z containing λx. Then

( )

(

)

x O Int Cl O

λ ∈ ⊆ , so there exist β-open sets V1 containing

λ

in

topolog-ical field  and V2 containing x in Z such that V V1⋅ 2⊆Int Cl O

(

( )

)

. Since

( )

x∈βCl A , then there is some b A V∈  2. Now

( ) (

2

) ( ) (

1 2

) ( )

(

( )

)

b A V A V V A Int Cl O

λ ∈ λ  λ ⊆ λ  ⋅ ⊆ λ  and hence

( )

x Clδ A

λ ∈ λ . Therefore λβCl A

( )

⊆Clδ

( )

λA .

Theorem 4.4 ForanysubsetAofanalmostβ-topologicalvectorspaceX, the followinghold:

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2) βCl A

( )

λ ⊆λCl Aδ

( )

for each non-zero scalar λ.

Proof. 1) Let y∈βCl x A

(

+

)

and let O be an open set in Z containing

x y

− + _{. Since}_Z_{is an almost}_β_{-topological vector space, there exist}_β_{-open sets}

1

V and V2 in Z such that − ∈x V1, y V∈ 2 and V V1+ 2 ⊆Int Cl O

(

( )

)

. Since

(

)

y∈βCl x A+ , there is some a∈

(

x A V+

)

 2 and hence

(

1 2

)

(

( )

)

x a A V V A Int Cl O

− + ∈ _ + ⊆ _ ⇒ − + ∈x y Cl Aδ

( )

⇒ ∈ +y x Cl Aδ

( )

.

Hence βCl x A

(

+

)

⊆ +x Cl Aδ

( )

.

2) Let x∈βCl A

( )

λ and O be an open set in Z containing _λ−1_x_{. So there}

exist β-open sets V1 in topological field  containing λ−1 and V2 in Z

containing x such that V V1⋅ 2 ⊆Int Cl O

(

( )

)

. As x∈βCl A

( )

λ ,

( )

λA V 2≠φ

and as a result, A Int Cl O_

(

( )

)

≠φ_{. Therefore} 1_{x Cl A}

( )

δ

λ− _∈ _{. Hence}

( )

Cl A Cl Aδ

β λ ⊆λ .

Theorem 4.5 LetAbeanopensetinanalmostβ-topologicalvectorspaceZ,

then:

1) βCl x A

(

+

)

⊆ +x Cl A

( )

_{for each} x E∈ _.

2) βCl A

( )

λ ⊆λCl A

( )

for each non zero scalar λ_.

Proof. 1) Let y∈βCl x A

(

+

)

and O be any open set in Z containing − +x y. Then there exist V V1, 2∈βO Z

( )

such that − ∈x V1, y V∈ 2 and

( )

(

)

1 2

V V+ ⊆Int Cl O _{. Since} y∈βCl x A

(

+

)

, there is some a∈

(

x A V+

)

 2.

Now − + ∈x a A V V

(

1+ 2

)

⊆A Int Cl O

(

( )

)

⇒A Int Cl O

(

( )

)

≠φ. Since A

is open, A O_ ≠φ _{. Thus}− + ∈x y Cl A

_{( )}

; that is, y x Cl A∈ +

( )

. Hence

(

)

( )

Cl x A x Cl A

β + ⊆ + .

2) Let x∈βCl A

( )

λ and O be any open set in Z containing _λ−1_y_{. Then}

there exist β-open sets V1 in topological field  containing λ−1 and V2 in

Z containing x such that V V1⋅ 2 ⊆Int Cl O

(

( )

)

. As x∈βCl A

( )

λ , there is some

( )

2

b∈ λA V . Thus _λ−1_{b A Int Cl O}_∈

(

( )

)

_⇒_{A Int Cl O}

(

( )

)

_≠_φ

  . Since A is

open, A O_ ≠φ_{. Thus} _λ−1_{y Cl A}_∈

( )

_{; that is,} _y_∈_λ_{Cl A}

_{( )}

_{. Hence}

( )

Cl A Cl A

β λ ⊆λ .

Theorem 4.6 LetAandBbesubsetsofanalmostβ-topologicalvectorspaceZ.

Then βCl A

( )

+βCl B

( )

⊆Cl A Bδ

(

+

)

.

Proof. Let x∈βCl A

( )

and y∈βCl B

( )

and let O be an open neighbor-hood of x y+ in Z. Since O Int Cl O⊆

(

( )

)

and Int Cl O

(

( )

)

is regular open, there exist V V1, 2∈βO Z

( )

such that x V∈ 1, y V∈ 2 and

( )

(

)

1 2

V V+ ⊆Int Cl O _{. Since} x∈βCl A

( )

and y∈βCl B

( )

, there are

1

a A V∈ _ _and b B V∈ _ ₂_{. Then}

(

) (

1 2

) (

)

(

( )

)

a b+ ∈ A B+ _ V V+ ⊆ A B+ _Int Cl O ⇒

(

A B+

)

_Int Cl O

(

( )

)

≠φ_.

Thus x y Cl A B+ ∈ δ

(

+

)

; that is, βCl A

( )

+βCl B

( )

⊆Cl A Bδ

(

+

)

.

Theorem 4.7 ForanysubsetAofanalmostβ-topologicalvectorspaceZ, the followingaretrue:

1) Int x Aδ

(

+

)

⊆ +x βInt A

( )

, and

2) x Int A+ δ

( )

⊆βInt x A

(

+

)

, for each x Z∈ .

Proof. 1) We need to show that for each y Int x A∈ δ

(

+

)

, − + ∈x y βInt A

( )

.

(8)

regular open set U in Z such that y U∈ ⊆Int x Aδ

(

+

)

. Since y Int x A∈ δ

(

+

)

,

y x a= + for some a A∈ _{. Since}_Z_{is almost}_β_{-topological vector space, then}

there exist β-open sets V1 and V2 in Z containing x and a respectively and

1 2

V V+ ⊆U _{. Thus}x V+ ₂⊆U ⇒V₂ ⊆ − + ⊆ − +x U x

(

x A+

)

=A. Since V2 is

β-open, then V2⊆βInt A

( )

and therefore a∈βInt A

( )

⇒ − + ∈x y βInt A

( )

y x βInt A

⇒ ∈ + . Hence the assertion follows.

2) Let y x Int A∈ + δ

( )

. Then there exists a regular open set U in Z such that

( )

x y U Int Aδ A

− + ∈ ⊆ ⊆ . By definition of almost β-topological vector spaces, we have β-open sets V1 and V2 in Z containing -x and y respectively, such

that V V1+ 2⊆U. Thus V2⊆ + ⊆ +x U x A ⇒ ∈y βInt x A

(

+

)

. Hence

( )

(

)

x Int A+ δ ⊆βInt x A+ .

Theorem 4.8 ForanysubsetAofanalmostβ-topologicalvectorspaceZ, the followingaretrue:

1) Intδ

( )

λA ⊆λβInt A

( )

, and

2) λInt Aδ

( )

⊆βInt A

( )

λ , for each non zero scalar λ.

Proof. Follows from the proof of above theorem by using second axiom of an almost β-topological vector space.

Theorem 4.9 LetZbeanalmostβ-topologicalvectorspace. Then

1) the translation mapping T Zx: →Z defined by T yx

( )

= + ∀x y x y Z, , ∈ , is almost β-continuous.

2) the multiplication mapping M Zλ: →Z defined by M xλ

( )

=λx x Z,∀ ∈ ,

is almost β-continuous, where λ_{be non-zero scalar in} __.

Proof. 1) Let y X∈ _{be an arbitrary. Let}_O_{be any open set in}_Z_containing

( )

x

T y . As O Int Cl O⊆

(

( )

)

, we have T yx

( )

∈Int Cl O

(

( )

)

Since Z is an almost β-topological vector space, there exist β-open sets V1 and V2 in Z containing

x and y respectively such that V V1+ 2 ⊆Int Cl O

(

( )

)

. Thus x V+ 2⊆Int Cl O

(

( )

)

( )

2

(

( )

)

x

T V Int Cl O

⇒ ⊆ . This proves that Tx is almost β-continuous at y. Since

y Z∈ was arbitrary, it follows that Tx is almost β-continuous.

2) Let x Z∈ _and_O_{be any open set in}_Z_containing M xλ

_{( )}

. Then there

ex-ist β-open sets V1 in the topological field  containing

λ

and V2 in Z

containing x such that V V1⋅ 2 ⊆Int Cl O

(

( )

)

. Thus λV2⊆Int Cl O

(

( )

)

( )

2

(

( )

)

M Vλ Int Cl O

⇒ ⊆ . This shows that Mλ is almost β-continuous at x and

hence Mλ is almost β-continuous everywhere in Z.

Theorem 4.10 For an almost β-topological vector space Z, the mapping

:Z Z Z

φ × → defined by φ

(

x y,

)

= +x y , ∀

(

x y,

)

∈ ×Z Z , is almost β-continuous.

Proof. Let

(

x y,

)

∈ ×Z Z and let U be regular open set in Z containing x y+ _.

Then, there exist β-open sets V1 and V2 in Z such that x V∈ 1, y V∈ 2 and

1 2

V V+ ⊆U _{. Since}V V₁× ₂_is_β_{-open in} Z Z× _{(with respect to product}

topol-ogy) such that

(

x y,

)

∈ ×V V1 2 and φ

(

V V1× 2

)

=V V1+ 2⊆U. It follows that φ

is almost β-continuous at

(

x y,

)

. Since

(

x y,

)

∈ ×Z Z is arbitrary, φ_{is almost}

β-continuous.

(9)

: Z Z

ψ _× → _defined _by φ λ

(

,x

)

=λx , ∀

(

λ,x

)

∈ ×_ Z , is almost β-continuous.

Proof. Follows from the proof of theorem 4.10 by using the second axiom of almost β-topological vector space.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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