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In this paper, we have introduced a new generalized form of topological vector 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 topological vector spaces. Some properties of β -topological vector spaces are also characterized.

The concept of topological vector spaces was introduced by Kolmogroff [

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 [

Definition 2.2 A subset A of a topological space X is said to be δ-open [

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

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 [

Also we recall some definitions that will be used later.

Definition 2.4 [

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 [

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 [

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 [

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