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 continuities 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, irresolute 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, almosts-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.

Throughout this paper, ( X , τ ) (or simply X) and ( Y , σ ) (or simply Y) mean topological spaces. For a subset A ⊆ X , C l ( A ) denotes the closure of A and I n t ( A ) denote the interior of A. The notation F denotes the field of real numbers ℝ or complex numbers ℂ with usual topology and ε , η represent the negligibly small positive numbers.

Definition 2.1 A subset A of a topological space X is said to be:

1) regular open if A = I n t ( C l ( A ) ) .

2) β-open [4] if A ⊆ C l ( I n t ( C l ( A ) ) ) .

Definition 2.2 A subset A of a topological space X is said to be δ-open [5] if for each 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 β I n t ( A ) (resp. I n t δ ( A ) ). 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 β I n t ( A ) . It is well known fact that a subset A ⊆ X is β-open (resp. δ-open) if and only if A = β I n t ( A ) (resp. A = I n t δ ( A ) ). 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 β C l ( A ) (resp. C l δ ( A ) ). It is also known that a subset A of X is β-closed (resp. δ-closed) if and only if A = β C l ( A ) (resp. A = C l δ ( A ) ). A point x ∈ β C l ( A ) if and only if A ∩ V ≠ ϕ for each β-open set V in X containing x. A point x ∈ C l δ ( A ) if A ∩ I n t ( C l ( 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 β O ( X ) (resp. β C ( X ) , R O ( X ) ). If A ∈ β O ( X ) , B ∈ β O ( Y ) , then A × B ∈ β O ( X × Y ) (with respect to the product topology). The family of all β-open sets in X containing x is denoted by β O ( X , x ) .

Definition 2.3 [11] A function f : X → Y from a topological space X to a topological space Y is called almost β-continuous at x ∈ X if for each open set O of Y containing f ( x ) , there exists V ∈ β O ( X , x ) such that f ( V ) ⊂ I n t ( C l ( O ) ) .

Also we recall some definitions that will be used later.

Definition 2.4 [12] Let T be a vector space over the field F . Let τ be a topology on T such that

1) For each x , y ∈ T and each open neighborhood O of x + y in T, there exist open neighborhoods O 1 and O 2 of x and y respectively in T such that O 1 + O 2 ⊆ O , and

2) For each λ ∈ F , x ∈ T and each open neighborhood O of λ x in T, there exists open neighborhoods O 1 of λ in F and O 2 of x in T such that O 1 O 2 ⊆ O .

Then the pair ( T ( F ) , τ ) is called topological vector space.

Definition 2.5 [8] Let T be a vector space over the field F . Let τ be a topology on T such that

1) For each x , y ∈ T and each open neighborhood O of x + y in T, there exist β-open sets V 1 and V 2 in T containing x and y respectively such that V 1 + V 2 ⊆ O , and

2) For each λ ∈ F , x ∈ T and each open neighborhood O of λ x in T, there exist β-open sets V 1 containing λ in F and V 2 containing x in T such that V 1 ⋅ V 2 ⊆ O .

Then the pair ( T ( F ) , τ ) is called β-topological vector space.

Definition 2.6 [13] Let T be a vector space over the field F . Let τ be a topology on T such that

1) For each x , y ∈ T and each regular open set U ⊆ T containing x + y , there exist pre-open sets P 1 and P 2 in T containing x and y respectively such that P 1 + P 2 ⊆ U , and

2) For each λ ∈ F , x ∈ T and each regular open set U ⊆ T containing λ x , there exist pre-open sets P 1 in F containing λ and P 2 containing x in T such that P 1 ⋅ P 2 ⊆ U .

Then the pair ( T ( F ) , τ ) is called an almost pretopological vector space.

Definition 2.7 [9] Let T be a vector space over the field F . Let τ be a topology on T such that

1) For each x , y ∈ T and each regular open set U ⊆ T containing x + y , there exist semi-open sets S 1 and S 2 in T containing x and y respectively such that S 1 + S 2 ⊆ U , and

2) For each λ ∈ F , x ∈ T and each regular open set U ⊆ T containing λ x , there exist semi-open sets S 1 in F containing λ and S 2 containing x in T such that S 1 ⋅ S 2 ⊆ U .

Then the pair ( T ( F ) , τ ) is called an almost s-topological vector space.

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 F ( ℝ or ℂ with standard topology). 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 V 1 and V 2 in Z containing x and y respectively such that V 1 + V 2 ⊆ U , and

2) For each λ ∈ F , x ∈ Z and each regular open set U ⊆ Z containing λ x , there exist β-open sets V 1 in F containing λ and V 2 containing x in Z such that V 1 V 2 ⊆ U .

Then the pair ( Z ( F ) , 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 = ℝ be the real vector space over the field F , where F = ℝ with the standard topology and τ be the usual topology endowed on Z, that is, τ is generated by the base B = { ( a , b ) : a , b ∈ ℝ } . Then ( Z ( F ) , τ ) is an almost β-topological vector space. For proving this, we have to verify the following two conditions:

1) Let x , y ∈ Z . Consider any regular open set U = ( x + y − ϵ , x + y + ϵ ) in Z containing x + y . Then we can opt for β-open sets V 1 = ( x − η , x + η ) and V 2 = ( y − η , y + η ) in Z containing x and y respectively, such that V 1 + V 2 ⊆ U

for each η < ϵ 2 . Thus first condition of the definition of almost β-topological vector space is satisfied.

2) Let λ ∈ F = ℝ and x ∈ Z . Consider a regular open set U = ( λ x − ϵ , λ x + ϵ ) in Z = ℝ containing λ x . Then we have the following cases:

Case (I). If λ > 0 and x > 0 , then λ x > 0 . We can choose β-open sets V 1 = ( λ − η , λ + η ) in F containing λ and V 2 = ( y − η , y + η ) in Z containing x, such that V 1 ⋅ V 2 ⊆ U for each η < ϵ λ + x + 1 .

Case (II). If λ < 0 and x < 0 , then λ x > 0 . We can choose β-open sets V 1 = ( λ − η , λ + η ) in F containing λ and V 2 = ( x − η , x + η ) in Z containing x, such that V 1 ⋅ V 2 ⊆ U for each η < ϵ 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 V 1 = ( λ − η , λ + η ) in F containing λ and V 2 = ( x − η , x + η ) in Z containing x, such that V 1 ⋅ V 2 ⊆ U for each

η < ϵ 1 + λ − x (resp. η < ϵ 1 − λ + x ).

Case (IV). If λ = 0 and x > 0 (resp. λ > 0 and x = 0 ), then λ x = 0 . We can select β-open neighborhoods V 1 = ( − η , η ) (resp. V 1 = ( λ − η , λ + η ) ) in F containing λ and V 2 = ( x − η , x + η ) (resp. V 2 = ( − η , η ) in Z containing x, such that V 1 ⋅ V 2 ⊆ U for each η < ϵ x + 1 (resp. η < ϵ 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 V 1 = ( − η , η ) (resp. V 1 = ( λ − η , λ + η ) ) in F containing λ and V 2 = ( x − η , x + η ) (resp. V 2 = ( − η , η ) in Z containing x, such that V 1 ⋅ V 2 ⊆ U for each η < ϵ 1 − x (resp. η < ϵ 1 − λ ).

Case (VI). If λ = 0 and x = 0 , then λ x = 0 . Then for β-open neighborhoods V 1 = ( − η , η ) of λ in F and V 2 = ( − η , η ) of x in Z, we have V 1 ⋅ V 2 ⊆ U for each η < ϵ .

This verifies the second condition of the definition of almost β-topological vector space.

Example 3.2 Let Z = ℝ be the real vector space over the field F with the topology τ generated by the base

B = { ( a , b ) : a , b ∈ ℝ } ∪ { ( c , d ) ∩ ℚ c : c , d ∈ ℝ } , where ℚ c denotes the set of irrational numbers. Then ( Z ( ℝ ) , τ ) is an almost β-topological vector space.

Example 3.3 Consider the field F = ℝ with standard topology. Let Z = ℝ be the real vector space over the field F endowed with topology τ = { ϕ , { 0 } , ℝ } . Then ( Z ( ℝ ) , 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 ∈ ℝ with 0 < c < d . Let Z = ℝ be the real vector space over the field F endowed with topology τ , where F = ℝ with the standard topology. Then ( Z ( F ) , 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 B = { [ a , b ) : a , b ∈ ℝ } and let this topology τ is imposed on the real vector space Z = ℝ over the topological field F = ℝ with standard topology. Then ( Z ( F ) , τ ) fails to be an almost β-topological vector space. For, U = [ 0,1 ) is regular open set in Z containing 0 = − 1.0 ( − 1 ∈ F = ℝ and 0 ∈ Z ) but there do not exist β-open sets V 1 in F containing −1 and V 2 in Z containing 0 such that V 1 ⋅ V 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 topological 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.

Throughout this section, an almost β-topological vector space ( Z ( F ) , τ ) over the topological field F will be simply written by Z and by a scalar, we mean an element from the topological field F .

Theorem 4.1 Let A be any δ-open set in an almost β-topological vector space Z. Then x + A , λ A ∈ β O ( Z ) , for each 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 V 1 and V 2 in Z such that − x ∈ V 1 , y ∈ V 2 such that V 1 + V 2 ⊆ U . Now − x + y ∈ − x + V 2 ⊆ U ⊆ A ⇒ V 2 ⊆ x + A . Since V 2 is β-open, y ∈ β I n t ( x + A ) . This shows that x + A = β I n t ( x + A ) . Hence x + A ∈ β O ( Z ) .

Further, let x ∈ λ A be arbitrary. Since A is δ-open, there exists a regular open set U in Z such that λ − 1 x ∈ U ⊆ A . Since Z is an almost β-topological vector space, there exist β-open sets V 1 in the topological field F containing λ − 1 and V 2 in Z containing x such that V 1 V 2 ⊆ U . Now λ − 1 x ∈ λ − 1 V 2 ⊆ A ⇒ V 2 ⊆ λ A ⇒ x ∈ β I n t ( λ A ) and hence λ A = β I n t ( λ A ) . Thus λ A is β-open in Z; i.e., λ A ∈ β O ( Z ) .

Theorem 4.2 Let B be any δ-closed set in an almost β-topological vector space Z. Then x + B , λ B ∈ β C l ( Z ) for each x ∈ Z and each non-zero scalar λ .

Proof. We need to show that x + B = β C l ( x + B ) . For, let y ∈ β C l ( 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 V 1 and V 2 in Z such that − x ∈ V 1 , y ∈ V 2 and V 1 + V 2 ⊆ U . Since y ∈ β C l ( x + B ) , then by definition, ( x + B ) ∩ V 2 ≠ ϕ ⇒ there is some a ∈ ( x + B ) ∩ V 2

⇒ − x + a ∈ B ∩ ( − x + V 2 ) ⊆ B ∩ ( V 1 + V 2 ) ⊆ B ∩ U ⊆ B ∩ W ⇒ B ∩ W ≠ ϕ . Thus − x + y ∈ C l δ ( B ) . Since B is δ-closed set, we have, − x + y ∈ B ⇒ y ∈ x + B . Therefore x + B = β C l ( x + B ) . Hence x + B ∈ β C l ( Z ) .

Next, we have to prove that λ B = β C l ( λ B ) . For, let x ∈ β C l ( λ B ) be arbitrary 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 V 1 containing λ − 1 in topological field F and V 2 containing x in Z such that V 1 ⋅ V 2 ⊆ U . Since x ∈ β C l ( λ B ) , then there is some a ∈ ( λ B ) ∩ V 2 . Now λ − 1 a ∈ B ∩ ( λ − 1 ⋅ V 2 ) ⊆ B ∩ ( V 1 ⋅ V 2 ) ⊆ B ∩ U ⊆ B ∩ W ⇒ B ∩ W ≠ ϕ . Thus λ − 1 x ∈ C l δ ( B ) = B ⇒ x ∈ λ B . Therefore λ B = β C l ( λ B ) . Hence

λ B ∈ β C l ( Z ) .

Theorem 4.3 For any subset A of an almost β-topological vector space Z, the following assertions hold:

1) x + β C l ( A ) ⊆ C l δ ( x + A ) for each x ∈ Z .

2) λ β C l ( A ) ⊆ C l δ ( λ A ) for each non zero scalar λ .

Proof. 1) Let z ∈ x + β C l ( A ) . Then z = x + y for some y ∈ β C l ( A ) . Let O be an open set in Z containing z, then z ∈ O ⊆ I n t ( C l ( O ) ) . Since Z is an almost β-topological vector space, then there exist V 1 , V 2 ∈ β O ( Z ) containing x and y respectively such that V 1 + V 2 ⊆ I n t ( C l ( O ) ) . Since y ∈ β C l ( A ) , then there is some a ∈ A ∩ V 2 . As a result, x + a ∈ ( x + A ) ∩ ( V 1 + V 2 ) ⊆ ( x + A ) ∩ I n t ( C l ( O ) ) ⇒ ( x + A ) ∩ I n t ( C l ( O ) ) ≠ ϕ . Thus z ∈ C l δ ( x + A ) . Therefore

x + β C l ( A ) ⊆ C l δ ( x + A ) .

2) Let x ∈ β C l ( A ) and let W be an open set in Z containing λ x . Then λ x ∈ O ⊆ I n t ( C l ( O ) ) , so there exist β-open sets V 1 containing λ in topological field F and V 2 containing x in Z such that V 1 ⋅ V 2 ⊆ I n t ( C l ( O ) ) . Since x ∈ β C l ( A ) , then there is some b ∈ A ∩ V 2 . Now

λ b ∈ ( λ A ) ∩ ( λ V 2 ) ⊆ ( λ A ) ∩ ( V 1 ⋅ V 2 ) ⊆ ( λ A ) ∩ I n t ( C l ( O ) ) and hence

λ x ∈ C l δ ( λ A ) . Therefore λ β C l ( A ) ⊆ C l δ ( λ A ) .

Theorem 4.4 For any subset A of an almost β-topological vector space X, the following hold:

1) β C l ( x + A ) ⊆ x + C l δ ( A ) for each x ∈ X .

2) β C l ( λ A ) ⊆ λ C l δ ( A ) for each non-zero scalar λ .

Proof. 1) Let y ∈ β C l ( 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 V 1 and V 2 in Z such that − x ∈ V 1 , y ∈ V 2 and V 1 + V 2 ⊆ I n t ( C l ( O ) ) . Since y ∈ β C l ( x + A ) , there is some a ∈ ( x + A ) ∩ V 2 and hence

− x + a ∈ A ∩ ( V 1 + V 2 ) ⊆ A ∩ I n t ( C l ( O ) ) ⇒ − x + y ∈ C l δ ( A ) ⇒ y ∈ x + C l δ ( A ) . Hence β C l ( x + A ) ⊆ x + C l δ ( A ) .

2) Let x ∈ β C l ( λ A ) and O be an open set in Z containing λ − 1 x . So there exist β-open sets V 1 in topological field F containing λ − 1 and V 2 in Z containing x such that V 1 ⋅ V 2 ⊆ I n t ( C l ( O ) ) . As x ∈ β C l ( λ A ) , ( λ A ) ∩ V 2 ≠ ϕ and as a result, A ∩ I n t ( C l ( O ) ) ≠ ϕ . Therefore λ − 1 x ∈ C l δ ( A ) . Hence β C l ( λ A ) ⊆ λ C l δ ( A ) .

Theorem 4.5 Let A be an open set in an almost β-topological vector space Z, then:

1) β C l ( x + A ) ⊆ x + C l ( A ) for each x ∈ E .

2) β C l ( λ A ) ⊆ λ C l ( A ) for each non zero scalar λ .

Proof. 1) Let y ∈ β C l ( x + A ) and O be any open set in Z containing − x + y . Then there exist V 1 , V 2 ∈ β O ( Z ) such that − x ∈ V 1 , y ∈ V 2 and V 1 + V 2 ⊆ I n t ( C l ( O ) ) . Since y ∈ β C l ( x + A ) , there is some a ∈ ( x + A ) ∩ V 2 . Now − x + a ∈ A ∩ ( V 1 + V 2 ) ⊆ A ∩ I n t ( C l ( O ) ) ⇒ A ∩ I n t ( C l ( O ) ) ≠ ϕ . Since A is open, A ∩ O ≠ ϕ . Thus − x + y ∈ C l ( A ) ; that is, y ∈ x + C l ( A ) . Hence β C l ( x + A ) ⊆ x + C l ( A ) .

2) Let x ∈ β C l ( λ A ) and O be any open set in Z containing λ − 1 y . Then there exist β-open sets V 1 in topological field F containing λ − 1 and V 2 in Z containing x such that V 1 ⋅ V 2 ⊆ I n t ( C l ( O ) ) . As x ∈ β C l ( λ A ) , there is some b ∈ ( λ A ) ∩ V 2 . Thus λ − 1 b ∈ A ∩ I n t ( C l ( O ) ) ⇒ A ∩ I n t ( C l ( O ) ) ≠ ϕ . Since A is open, A ∩ O ≠ ϕ . Thus λ − 1 y ∈ C l ( A ) ; that is, y ∈ λ C l ( A ) . Hence β C l ( λ A ) ⊆ λ C l ( A ) .

Theorem 4.6 Let A and B be subsets of an almost β-topological vector space Z. Then β C l ( A ) + β C l ( B ) ⊆ C l δ ( A + B ) .

Proof. Let x ∈ β C l ( A ) and y ∈ β C l ( B ) and let O be an open neighborhood of x + y in Z. Since O ⊆ I n t ( C l ( O ) ) and I n t ( C l ( O ) ) is regular open, there exist V 1 , V 2 ∈ β O ( Z ) such that x ∈ V 1 , y ∈ V 2 and

V 1 + V 2 ⊆ I n t ( C l ( O ) ) . Since x ∈ β C l ( A ) and y ∈ β C l ( B ) , there are

a ∈ A ∩ V 1 and b ∈ B ∩ V 2 . Then

a + b ∈ ( A + B ) ∩ ( V 1 + V 2 ) ⊆ ( A + B ) ∩ I n t ( C l ( O ) ) ⇒ ( A + B ) ∩ I n t ( C l ( O ) ) ≠ ϕ . Thus x + y ∈ C l δ ( A + B ) ; that is, β C l ( A ) + β C l ( B ) ⊆ C l δ ( A + B ) .

Theorem 4.7 For any subset A of an almost β-topological vector space Z, the following are true:

1) I n t δ ( x + A ) ⊆ x + β I n t ( A ) , and

2) x + I n t δ ( A ) ⊆ β I n t ( x + A ) , for each x ∈ Z .

Proof. 1) We need to show that for each y ∈ I n t δ ( x + A ) , − x + y ∈ β I n t ( A ) . We know I n t δ ( x + A ) is δ-open. Then for each y ∈ I n t δ ( x + A ) , there exists a regular open set U in Z such that y ∈ U ⊆ I n t δ ( x + A ) . Since y ∈ I n t δ ( x + A ) , y = x + a for some a ∈ A . Since Z is almost β-topological vector space, then there exist β-open sets V 1 and V 2 in Z containing x and a respectively and V 1 + V 2 ⊆ U . Thus x + V 2 ⊆ U ⇒ V 2 ⊆ − x + U ⊆ − x + ( x + A ) = A . Since V 2 is β-open, then V 2 ⊆ β I n t ( A ) and therefore a ∈ β I n t ( A ) ⇒ − x + y ∈ β I n t ( A ) ⇒ y ∈ x + β I n t ( A ) . Hence the assertion follows.

2) Let y ∈ x + I n t δ ( A ) . Then there exists a regular open set U in Z such that − x + y ∈ U ⊆ I n t δ ( A ) ⊆ A . By definition of almost β-topological vector spaces, we have β-open sets V 1 and V 2 in Z containing -x and y respectively, such that V 1 + V 2 ⊆ U . Thus V 2 ⊆ x + U ⊆ x + A ⇒ y ∈ β I n t ( x + A ) . Hence x + I n t δ ( A ) ⊆ β I n t ( x + A ) .

Theorem 4.8 For any subset A of an almost β-topological vector space Z, the following are true:

1) I n t δ ( λ A ) ⊆ λ β I n t ( A ) , and

2) λ I n t δ ( A ) ⊆ β I n t ( λ A ) , 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 Let Z be an almost β-topological vector space. Then

1) the translation mapping T x : Z → Z defined by T x ( y ) = x + y , ∀ x , y ∈ Z , is almost β-continuous.

2) the multiplication mapping M λ : Z → Z defined by M λ ( x ) = λ x , ∀ x ∈ Z , is almost β-continuous, where λ be non-zero scalar in F .

Proof. 1) Let y ∈ X be an arbitrary. Let O be any open set in Z containing T x ( y ) . As O ⊆ I n t ( C l ( O ) ) , we have T x ( y ) ∈ I n t ( C l ( O ) ) Since Z is an almost β-topological vector space, there exist β-open sets V 1 and V 2 in Z containing x and y respectively such that V 1 + V 2 ⊆ I n t ( C l ( O ) ) . Thus x + V 2 ⊆ I n t ( C l ( O ) ) ⇒ T x ( V 2 ) ⊆ I n t ( C l ( O ) ) . This proves that T x is almost β-continuous at y. Since y ∈ Z was arbitrary, it follows that T x is almost β-continuous.

2) Let x ∈ Z and O be any open set in Z containing M λ ( x ) . Then there exist β-open sets V 1 in the topological field F containing λ and V 2 in Z containing x such that V 1 ⋅ V 2 ⊆ I n t ( C l ( O ) ) . Thus λ V 2 ⊆ I n t ( C l ( O ) ) ⇒ M λ ( V 2 ) ⊆ I n t ( C l ( O ) ) . This shows that M λ is almost β-continuous at x and hence M λ is almost β-continuous everywhere in Z.

Theorem 4.10 For an almost β-topological vector space Z, the mapping ϕ : Z × Z → Z 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 V 1 and V 2 in Z such that x ∈ V 1 , y ∈ V 2 and V 1 + V 2 ⊆ U . Since V 1 × V 2 is β-open in Z × Z (with respect to product topology) such that ( x , y ) ∈ V 1 × V 2 and ϕ ( V 1 × V 2 ) = V 1 + V 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.

Theorem 4.11 For an almost β-topological vector space Z, the mapping ψ : F × Z → Z defined by ϕ ( λ , x ) = λ x , ∀ ( λ , x ) ∈ F × 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.

The authors declare no conflicts of interest regarding the publication of 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