Journal of Advanced Studies in Topology 9:2 (2018), 139–146

On almost s-topological vector spaces

a, a a a Madhu Ram ∗, Shallu Sharma , Sahil Billawria , Amjad Hussain aDepartment of , University of Jammu, JK-180006, India.

Abstract In this paper, we introduce the notion of almost s-topological vector spaces and present some examples and counterexamples of almost s-topological vector spaces. Some general properties of almost s-topological vector spaces are investigated. Keywords: Semi-open sets, regular open sets, almost s-topological vector spaces. 2010 MSC: 57N17, 57N99.

1. Introduction

The concept of topological vector spaces was first given by Kolmogroff [4] in 1934. Since then, different researchers have been explored many interesting and useful properties of topological vector spaces. Due to nice properties, topological vector spaces earn a great importance and remain a fundamental notion in fixed point theory, , variational inequalities, etc. Nowadays the researchers not only make use of topological vector spaces in other fields for developing new notions but also stretch and extend the concept of topological vector spaces with every possible way, making this field of study a more convenient and understandable. Recently, M. Khan et al. [2] introduced the s-topological vector spaces; a generalization of topological vector spaces. In 2016, Khan and Iqbal [3] introduced the irresolute topological vector spaces which are a particular brand of s-topological vector spaces but irresolute topological vector spaces are independent of topological vector spaces. Ibrahim [1] initiated the study of α topological vector spaces. − S. Sharma et. al. [7] put forth the concept of almost pretopological vector spaces. The aim of this paper is to introduce the notion of almost s-topological vector spaces and present some examples of almost s- topological vector spaces. Subsequently, some general properties of almost s-topological vector spaces are investigated.

2. Preliminaries

Throughout this paper, (X, τ) (or simply X) means a topological . For a subset A X, the of A ⊆ and the of A are denoted by Cl(A) and Int(A) respectively. The notation R ( resp. C) represents the

∗Corresponding author Email address: [email protected] (Madhu Ram ) Received: 22 October 2018 Accepted: 23 November 2018 http://dx.doi.org/10.20454/jast.2018.1490 2090-8288 c 2018 Modern Science Publishers. All rights reserved.

M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 140 set of real numbers (resp. the set of complex numbers) and  (resp. η) represents negligibly small positive number.

Definition 2.1. Let X be a . A subset A of X is called: (a) regular open if A = Int(Cl(A)). (b) semi-open [5] if A Cl(Int(A)). ⊆ Definition 2.2. A subset A of a topological space X is said to be δ open [9] if for each x A, there exists a − ∈ regular open set U in X such that x U A. ∈ ⊆ The complement of a regular open (resp. semi-open, δ open) set is called regular closed (resp. semi-closed, − δ closed [9]). The intersection of all semi-closed (resp. δ closed) sets in X containing a subset A X is − − ⊆ called the semi-closure (resp. δ closure) of A and is denoted by sCl(A) (resp. Cl (A)). It is known that a − δ subset A of X is semi-closed (resp. δ closed) if and only if A = sCl(A) (resp. A = Cl (A)). A point x Cl (A) − δ ∈ δ if and only if A Int(Cl(U)) , for each open set U in X containing x. The union of all semi-open (resp. ∩ ∅ δ open) sets in X that are contained in A X is called the semi-interior (resp. δ interior) of A and is − ⊆ − denoted by sInt(A) (resp. Int (A)). A point x X is called a semi-interior point of A X if there exists a δ ∈ ⊆ semi-open set U in X such that x U A. The set of all semi-interior points of A is equal to sInt(A). It is ∈ ⊆ well-known that a subset A X is semi-open (resp. δ open) if and only if A = sInt(A) (resp. A = Int (A)). ⊆ − δ The family of all regular open (resp. semi-open, semi-closed) sets in X is denoted by RO(X) (resp. SO(X), SC(X)). The family of all semi-open sets in X containing x is denoted by SO(X, x). If A SO(X), B SO(Y) ∈ ∈ (X and Y are topological spaces), then A B SO(X Y) (with respect to the ). It is very × ∈ × well known that the family of all δ open sets in X forms a topology on X, denoted by τ and the class of all − δ regular open sets in X forms an open for τδ. Recall that a subset A of X is called semi-neighborhood of a point x of X if there exists a semi-open set U in X containing x such that U A. If A itself is semi-open, then A is called semi-open neighborhood of x. ⊆ Definition 2.3. [6] A f : X Y from a topological space X to a topological space Y is called almost → semi-continuous if for each x X and each regular open set V Y containing f (x), there exists a semi-open ∈ ⊆ set U X containing x such that f (U) V. ⊆ ⊆ Also, we recall some definitions that will be used later.

Definition 2.4. Let L be a over the field F (R or C). Let τ be a topology on L such that (1) For each x, y L and each open neighborhood W of x + y in L, there exist open neighborhoods U and V ∈ of x and y respectively, in L such that U + V W and ⊆ (2) For each λ F, x L and each open neighborhood W of λx in L, there exist open neighborhoods U of λ ∈ ∈ in F and V of x in L such that U.V W. ⊆ Then the pair (L(F), τ) is called topological vector space. Definition 2.5. [2] Let L be a vector space over the field F (R or C) and let τ be a topology on L such that (1) For each x, y L and each open set W in L containing x + y, there exist semi-open sets U and V in L ∈ containing x and y respectively such that U + V W and ⊆ (2) For each λ F, x L and each open set W in L containing λx, there exist semi-open sets U in F containing ∈ ∈ λ and V in L containing x such that U.V W. ⊆ Then the pair (L(F), τ) is called s-topological vector space. M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 141

Definition 2.6. [3] Let L be a vector space over the field F (R or C) and τ be a topology on L such that (1) For each x, y L and each semi-open set W in L containing x + y, there exist semi-open sets U and V in ∈ L containing x and y respectively such that U + V W and ⊆ (2) For each λ F, x L and each semi-open set W in L containing λx, there exist semi-open sets U in F ∈ ∈ containing λ and V in L containing x such that U.V W. ⊆ Then the pair (L(F), τ) is called irresolute topological vector space.

3. Almost s-topological vector spaces

In this section, we define almost s-topological vector spaces and investigate their relationships with certain other types of spaces.

Definition 3.1. Let E be a vector space over the field K, where K = R or C with standard topology. Let τ be a topology on E such that the following axioms are satisfied: (1) For each x, y E and each regular open set W E containing x + y, there exist semi-open sets U and V ∈ ⊆ in E containing x and y respectively, such that U + V W, and ⊆ (2) For each λ K, x E and each regular open set W E containing λx, there exist semi-open sets U in K ∈ ∈ ⊆ containing λ and V in E containing x such that U.V W. ⊆ Then the pair (E(K), τ) is called an almost s-topological vector space (written in short, ASTVS).

Let us particularize some examples of almost s-topological vector spaces.

Example 3.2. Consider the topological field K = R with standard topology. Let E = R be endowed with the topology τ on E generated by the base B = (a, b): a, b R . Then (E , τ) is an almost s-topological vector { ∈ } (K) space. Example 3.3. Let E = R be the vector space of real numbers over the field K, where K = R with standard topology and the topology τ on E be generated by the base = (a, b), [c, d): a, b, c and d are real numbers B { with 0 < c < d . We show that (E , τ) is an almost s-topological vector space. For which we have to verify } (K) the following two conditions: (1) Let x, y E. Then, for regularly open set W = [x + y, x + y + ) (resp. (x + y , x + y + )) in E, we can ∈ − opt for semi-open neighborhoods U = [x, x + η) (resp. (x η, x + η)) and V = [y, y + η) (resp. (y η, y + η)) − − of x and y respectively in E such that U + V W for each η <  . ⊆ 2 (2) Let x E and λ K. We have following cases: ∈ ∈ Case (I). If λ > 0 and x > 0. Consider any regularly open set W = [λx, λx + ) (resp. (λx , λx + )) in E − containing λx. We can choose semi-open neighborhoods U = [λ, λ + η) (resp. (λ η, λ + η)) of λ in K and − V = [x, x + η) (resp. (x η, x + η)) of x in E such that U.V W for each η <  . − ⊆ λ+x+1 Case (II). If λ < 0 and x < 0, then λx > 0. So, for regularly open set W = [λx, λx + ) (resp. (λx , λx + )) − in E containing λx, we can choose semi-open neighborhoods U = (λ η, λ] (resp. (λ η, λ + η)) of λ in K − −  and V = (x η, x] (resp. (x η, x + η)) of x in E such that U.V W for sufficiently appropriate η λ+−x 1 . − − ⊆ ≤ − Case (III). If λ = 0 and x > 0 ( resp. λ > 0 and x = 0). Then λx = 0. Therefore, for any regularly open set W = ( , ) in E, we can opt for semi-open neighborhoods U = ( η, η) (resp. U = (λ η, λ + η)) in K = R − − − and V = (x η, x + η) (resp. V = ( η, η)) in E such that U.V W for each η <  (resp. η <  ). − − ⊆ x+1 λ+1 Case (IV). If λ = 0 and x < 0 (resp. λ < 0 and x = 0). Then, for the selection of semi-open neighborhoods U = ( η, η) (resp. U = (λ η, λ + η)) of λ in R and V = (x η, x + η) (resp. V = ( η, η)) of x in E, we have − −   − − U.V W = ( , ) for each η < 1 x (resp. (η < 1 λ )). ⊆ − − − M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 142

Case (V). If λ = 0 and x = 0. Consider any regularly open set W = ( , ) in E containing 0 = 0.0, we can − find semi-open sets U = ( η, η) in R and V = ( η, η) in E such that U.V W for each η < √. Thus, (E R , τ) − − ⊆ ( ) is an almost s-topological vector space.

It is obvious from the definition that every s-topological vector space is an almost s-topological vector space but the converse is not true in general. The following are some examples of almost s-topological vector spaces which are not s-topological vector spaces:

Example 3.4. Consider the field K = R with standard topology. Let E = R be the vector space over the field K, where E is endowed with the topology τ = , 0 , R . Then (E R , τ) is an almost s-topological vector {∅ { } } ( ) space.

Example 3.5. Let E = R be the real vector space with the topology τ on E generated by the base B = (a, b): { a, b R (c, d) D : c, d R and D denotes the set of irrational numbers . We show that (E R , τ) is an ∈ } ∪ { ∩ ∈ } ( ) almost s-topological vector space. For this, we have to convince the following conditions are valid: (1) Let x, y X. Consider any regular open set W = (x + y , x + y + ) of E containing x + y. Then we can ∈ − choose semi-open sets U = (x η, x + η) and V = (y η, y + η) in E containing x and y respectively, such that − − U + V W for each η <  . This verifies the first condition of the definition of almost s-topological vector ⊆ 2 spaces. (2) Let λ R and x E. Consider a regular open set W = (λx , λx + ) in E containing λx. We have ∈ ∈ − following cases: Case (I). If λ > 0 and x > 0, then for the choice of semi-open sets U = (λ η, λ + η) in R containing λ and − V = (x η, x + η) in E containing x, we have U.V W for each η <  . − ⊆ λ+x+1 Case (II). If λ < 0 and x < 0, then λx > 0. We can choose semi-open neighborhoods U = (λ η, λ + η) of λ  − in R and V = (x η, x + η) of x in E such that U.V W for sufficiently appropriate η λ+−x 1 . − ⊆ ≤ − Case (III). If λ = 0 and x > 0 ( resp. λ > 0 and x = 0). Then, for the selection of semi-open neighborhoods U = ( η, η) (resp. U = (λ η, λ + η)) of λ in R and V = (x η, x + η) (resp. V = ( η, η)) of x in E, we see that − − − − U.V W for each η <  (resp. η <  ). ⊆ x+1 λ+1 Case (IV). If λ = 0 and x < 0 (resp. λ < 0 and x = 0). Then, for the selection of semi-open sets U = ( η, η) − (resp. U = (λ η, λ + η)) in R containing λ and V = (x η, x + η) (resp. V = ( η, η)) in E containing x, we −   − − have U.V W for every η < 1 x (resp. η < 1 λ ). ⊆ − − Case (V). If λ = 0 and x = 0. Then, for semi-open neighborhoods U = ( η, η) of λ in R and V = ( η, η) of x − − in E, we have U.V W for each η < √. ⊆ Thus, both conditions of the definition of almost s-topological vector spaces are verified. Hence (E(R), τ) is an almost s-topological vector space.

So far we have presented examples of almost s-topological vector spaces. We present now some examples which fall beyond the class of almost s-topological vector spaces.

Example 3.6. Consider the field K = R with standard topology. Let E = R be endowed with the topology τ generated by the family of sets [a, b): a, b R . Then (E R , τ) is not almost s-topological vector space { ∈ } ( ) because for regularly open set W = [0, 1) in E containing 0 = 1.0 ( 1 K = R and 0 E), there do not exist − − ∈ ∈ semi-open sets U in K containing 1 and V in E containing 0 such that U.V W. − ⊆ Example 3.7. Consider the topological field K = R with standard topology. Let E = R be given with discrete topology. Then (E(R), τ) is not almost s-topological vector space. M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 143

4. General properties

Throughout this section, an almost s-topological vector space E means an almost s-topological vector space (E(K), τ) and by a scalar, we mean an element of the topological field K. we turn now to the basic properties of almost s-topological vector spaces.

Theorem 4.1. Let A be any δ open set in an almost s-topological vector space E. Then the following are − true: (1) x + A SO(E) for each x E. ∈ ∈ (2) λA SO(E) for each non-zero scalar λ. ∈ Proof. (1) Let y x+A. Then y = x+a for some a A. Since A is δ open, there exists a regular open set W in ∈ ∈ − E such that a W A x + y W. Since E is an almost s-topological vector space, there exist semi-open ∈ ⊆ ⇒ − ∈ sets U and V in E such that x U, y V and U + V W U + V A x + V A V x + A. − ∈ ∈ ⊆ ⇒ ⊆ ⇒ − ⊆ ⇒ ⊆ Since V is semi-open, y sInt(x + A). This shows that x + A = sInt(x + A). Hence x + A SO(E). ∈ ∈ (2) Let x λA be an arbitrary. Since A is δ open, there exists a regular open set W in E such that 1 x W A. ∈ − λ ∈ ⊆ By definition of almost s-topological vector spaces, there exist semi-open sets U in the topological field K containing 1 and V in E containing x such that U.V W. This gives that V λA. Since V is semi-open, λ ⊆ ⊆ x sInt(λA) and hence λA = sInt(λA). This reflects that λA is semi-open in E; that is, λA SO(E). ∈ ∈ Corollary 4.2. For any δ open set A in an almost s-topological vector space E, the following are valid: − (1) x + A Cl(Int(x + A)) for each x E. ⊆ ∈ (2) λA Cl(Int(λA)) for each non-zero scalar λ. ⊆ Theorem 4.3. Let A be any δ open set in an almost s-topological vector space E. Then A + B SO(E) for − ∈ any subset B E. ⊆ Proof. Follows directly from Theorem 4.1.

Theorem 4.4. Let B be any δ in an almost s-topological vector space E. Then the following are − true: (1) x + B SC(E) for each x E. ∈ ∈ (2) λB SC(E) for each non-zero scalar λ. ∈ Proof. (1) Let y sCl(x + B) be an arbitrary. Fix z = x + y. Let W be any open set in E containing z. Then ∈ − there exist U, V SO(E) such that x U, y V and U + V Int(Cl(W)). By assumption, (x + B) V , ∈ − ∈ ∈ ⊆ ∩ ∅ ⇒ there is a (x + B) V x + a B (U + V) B Int(Cl(W)) B Int(Cl(W)) , z Cl (B). But B ∈ ∩ ⇒ − ∈ ∩ ⊆ ∩ ⇒ ∩ ∅ ⇒ ∈ δ is δ closed, y x + B. Therefore x + B = sCl(x + B). Hence x + B SC(E). − ∈ ∈ (2) In order to show that λB SC(E), it suffices to show that λB = sCl(λB). So, let x sCl(λB) and let W be ∈ ∈ any open set in E containing 1 x. Since Int(Cl(W)) is regularly open in E such that W Int(Cl(W)), there exist λ ⊆ semi-open sets U in the topological field K containing 1 and V in E containing x such that U.V Int(Cl(W)). λ ⊆ By assumption, there is a (λB) V and whence we find that B Int(Cl(W)) , 1 x Cl (B) = B x λB ∈ ∩ ∩ ∅ ⇒ λ ∈ δ ⇒ ∈ showing that λB = sCl(λB). Hence the proof is finished.

Corollary 4.5. Let A be any δ closed set in an almost s-topological vector space E. Then − (1) Int(Cl(x + A)) x + A for each x E. ⊆ ∈ (2) Int(Cl(λA)) λA for each non-zero scalar λ. ⊆ Theorem 4.6. For any subset A of an almost s-topological vector space E, the following are valid: (1) x + sCl(A) Cl (x + A) for each x E. ⊆ δ ∈ (2) λsCl(A) Cl (λA) for each non-zero scalar λ. ⊆ δ M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 144

Proof. (1) Suppose that z x + sCl(A). Then z = x + y for some y sCl(A). Let W be an open set in ∈ ∈ E containing z. Since W Int(Cl(W)), there exist U, V SO(E) containing x and y respectively such that ⊆ ∈ U + V Int(Cl(W)). Since y sCl(A), A V , . Therefore, there is a A V and as a result, x + a ⊆ ∈ ∩ ∅ ∈ ∩ ∈ (x+A) Int(Cl(W)) (x+A) Int(Cl(W)) , . Thus, z Cl (x+A). This reflects that x+sCl(A) Cl (x+A). ∩ ⇒ ∩ ∅ ∈ δ ⊆ δ (2) Suppose that x sCl(A). Let W be an open set in E containing λx. Then there exist semi-open sets U ∈ in K containing λ and V in E containing x such that U.V Int(Cl(W)). By assumption, A V , . So, ⊆ ∩ ∅ there is 1 A V. Now λ1 (λA) U.V (λA) Int(Cl(W)) (λA) Int(Cl(W)) , and consequently ∈ ∩ ∈ ∩ ⊆ ∩ ⇒ ∩ ∅ λx Cl (A). Therefore λsCl(A) Cl (λA). ∈ δ ⊆ δ If the set A in Theorem 4.4 is replaced by δ open set, we obtain the following result: − Theorem 4.7. Let A be any δ open set in an almost s-topological vector space E. Then − (1) x + sCl(A) Cl(x + A) for each x E. ⊆ ∈ (2) λsCl(A) Cl(λA) for each non-zero scalar λ. ⊆ Proof. (1) For any z x + sCl(A), z = x + y for some y sCl(A). Let W be any open set in E containing z. ∈ ∈ Since Int(Cl(W)) is regularly open in E containing z, there exist semi-open sets U in E containing x and V in E containing y such that U + V Int(Cl(W)). By assumption, there exists 1 A V. Whence we conclude ⊆ ∈ ∩ that (x + A) Int(Cl(W)) , . ∩ ∅ Since A is δ open, by Theorem 4.1, x + A is semi-open. Hence it follows that Cl(Int(x + A)) Int(Cl(W)) , − ∩ Int(x + A) Int(Cl(W)) , Int(x + A) Cl(W) , Int(x + A) W , (x + A) W , . Thus, ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ z Cl(x + A) and thereby x + sCl(A) Cl(x + A). ∈ ⊆ (2) If x sCl(A) and W be any open set in E containing λx, then there exist semi-open sets U in the topological ∈ field K containing λ and V in E containing x such that U.V Int(Cl(W)). The consideration y sCl(A) implies ⊆ ∈ that A V , . Consequently, there is a A V which implies that λa (λA) Int(Cl(W)). Therefore, ∩ ∅ ∈ ∩ ∈ ∩ (λA) Int(Cl(W)) , . By Theorem 4.1, λA is semi-open in E. Hence it follows that Cl(Int(λA)) Int(Cl(W)) , ∩ ∅ ∩ Int(λA) Int(Cl(W)) , Int(λA) Cl(W) , Int(λA) W , (λA) W , λx Cl(λA). ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ ⇒ ∩ ∅ ⇒ ∈ Thus λsCl(A) Cl(λA). ⊆ Theorem 4.8. For any subset A of an almost s-topological vector space E, the following are valid: (1) sCl(x + A) x + Cl (A) for each x E. ⊆ δ ∈ (2) sCl(λA) λCl (A) for each non-zero scalar λ. ⊆ δ Proof. (1) Let y sCl(x + A) and let W be any open set in E containing x + y. Since E is an almost ∈ − s-topological vector space and W Int(Cl(W)), there exist semi-open sets U and V in E with the property ⊆ that x U, y V and U + V Int(Cl(W)). Considering y sCl(x + A), there is a (x + A) V and hence − ∈ ∈ ⊆ ∈ ∈ ∩ x+a A (U +V) A Int(Cl(W)) A Int(Cl(W)) , x+ y Cl (A); hence sCl(x+A) x+Cl (A). − ∈ ∩ ⊆ ∩ ⇒ ∩ ∅ ⇒ − ∈ δ ⊆ δ 1 (2) It suffices to show that λ x Clδ(A) for any x sCl(λA). So, let x sCl(λA) and W be an open set 1 ∈ ∈ ∈ 1 in E containing λ x. Then there exist semi-open sets U in the topological field K containing λ and V in E containing x such that U.V Int(Cl(W)). As x sCl(λA), (λA) V , and as a result, A Int(Cl(W)) , . ⊆ ∈ ∩ ∅ ∩ ∅ Therefore 1 x Cl (A). Hence the assertion follows. λ ∈ δ If the set A in Theorem 4.4 is replaced by open set, we obtain the following result:

Theorem 4.9. Let A be an open set in an almost s-topological vector space E. The following assertions are valid: (1) sCl(x + A) x + Cl(A) for each x E. ⊆ ∈ (2) sCl(λA) λCl(A) for each non-zero scalar λ. ⊆ Proof. (1) We show that for each y sCl(x + A), x + y Cl(A) holds. Assume now that y sCl(x + A) and ∈ − ∈ ∈ let W be any open set in E containing x + y. Then there exist U, V SO(E) such that x U, y V and − ∈ − ∈ ∈ M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 145

U + V Int(Cl(W)). Considering y sCl(x + A), there is 1 (x + A) V and thus, x + 1 A Int(Cl(W)) ⊆ ∈ ∈ ∩ − ∈ ∩ ⇒ A Int(Cl(W)) , A Cl(W) , . Since A is open, A W , , shows that x + y Cl(A); that is, ∩ ∅ ⇒ ∩ ∅ ∩ ∅ − ∈ y x + Cl(A). Hence sCl(x + A) x + Cl(A). ∈ ⊆ (2) As in the proof of part (1), it follows that sCl(λA) λCl(A). ⊆ Theorem 4.10. Let A be any subset of an almost s-topological vector space E. Then (1) Int (x + A) x + sInt(A) for each x E. δ ⊆ ∈ (2) x + Int (A) sInt(x + A) for each x E. δ ⊆ ∈ Proof. (1) It suffices to show that for each y Int (x + A), x + y sInt(A). Since Int (x + A) is δ open, for ∈ δ − ∈ δ − each y Int (x+A), there exists a regular open set W in E such that y W Int (x+A). Since y Int (x+A), ∈ δ ∈ ⊆ δ ∈ δ y = x+a for some a A. By definition of almost s-topological vector spaces, there exist semi-open sets U and ∈ V in E containing x and a respectively, such that U + V W. This implies that V x + W x + x + A = A. ⊆ ⊆ − ⊆ − By Theorem 4.1, x + W is semi-open and thereby x + y x + W sInt(A); hence the assertion follows. − − ∈ − ⊆ (2) Let y x + Int (A). Then there exists a regular open set W in E such that x + y W A. By definition of ∈ δ − ∈ ⊆ almost s-topological vector spaces, we obtain semi-open sets U and V in E containing x and y respectively, − such that U + V W V x + W x + A. Since V is semi-open, V sInt(x + A); that is, y sInt(x + A). ⊆ ⇒ ⊆ ⊆ ⊆ ∈ Thus, x + Int (A) sInt(x + A). δ ⊆ The analog of Theorem 4.8 is the following: Theorem 4.11. Let A be any subset of an almost s-topological vector space E. Then the following are true: (1) Int (λA) λsInt(A) for each non-zero scalar λ. δ ⊆ (2) λInt (A) sInt(λA) for each non-zero scalar λ. δ ⊆ Theorem 4.12. Let A be any semi-open set in an almost s-topological vector space E. The following assertions are valid: (1) x + Int(A) sInt(x + A) for each x E. ⊆ ∈ (2) λInt(A) sInt(λA) for each non-zero scalar λ. ⊆ Proof. (1) Let y x + Int(A). Since A is semi-closed, Int(A) is regularly open set in E. Consequently, there ∈ exist U, V SO(E) such that x U, y V and U + V Int(A). This gives x + V A V x + A. Since ∈ − ∈ ∈ ⊆ − ⊆ ⇒ ⊆ V is semi-open, y sInt(x + A); hence x + Int(A) sInt(x + A). ∈ ⊆ (2) If x λInt(A), then 1 x Int(A). Since Int(A) is regularly open, we get semi-open sets U SO(K) and ∈ λ ∈ ∈ V SO(E) such that 1 U, x V and U.V Int(A). This implies that V λA V sInt(λA) because V is ∈ λ ∈ ∈ ⊆ ⊆ ⇒ ⊆ semi-open set in E. This show that x sInt(λA). In other words, λInt(A) sInt(λA). ∈ ⊆ Here we particularize properties of some special functions on almost s-topological vector spaces. Theorem 4.13. Let E be an almost s-topological vector space. Then

(1) the translation mapping Tx : E E defined by Tx(y) = x + y, x, y E, is almost semi-continuous. → ∀ ∈ (2) the multiplication mapping M : E E defined by M (x) = λx, x E (λ is non-zero fixed scalar), is λ → λ ∀ ∈ almost semi-continuous.

Proof. (1) Let y E be an arbitrary. Let W be any regular open set in E containing Tx(y). Then, by ∈ definition of almost s-topological vector spaces, there exist semi-open sets U in E containing x and V in E containing y such that U + V W. This results in x + V W Tx(V) W. This indicates that Tx is almost ⊆ ⊆ ⇒ ⊆ semi-continuous at y and hence Tx is almost semi-continuous. (2) Let x E and W be any regular open set in E containing λx. Then there exist semi-open sets U in the ∈ topological field K containing λ and V in E containing x such that U.V W. This gives that λV W. This ⊆ ⊆ means that M (V) W showing that M is almost semi-continuous at x. Since x E was an arbitrary, it λ ⊆ λ ∈ follows that Mλ is almost semi-continuous. M. Ram et al., Journal of Advanced Studies in Topology 9:2 (2018), 139–146 146

Theorem 4.14. For an almost s-topological vector space E, the mapping φ : E E E defined by φ(x, y) = × → x + y, (x, y) E E, is almost semi-continuous. ∀ ∈ × Proof. Let (x, y) E E and let W be any regular open set in E such that φ(x, y) E E. Then there exist ∈ × ∈ × semi-open set U and V in E such that x U, y V and U + V W. Since U V is semi-open in E E (with ∈ ∈ ⊆ × × respect to the product topology) such that (x, y) U V and φ(U V) = U + V W, it follows that φ is ∈ × × ⊆ almost semi-continuous at (x, y) and consequently, φ is almost semi-continuous.

Theorem 4.15. For an almost s-topological vector space E, the mapping ψ : K E E defined by ψ(λ, x) = λx, × → (λ, x) K E, is almost semi-continuous. ∀ ∈ × Proof. Let λ K, x E be arbitrary. Let W be any regular open set in E containing λx. Then, there ∈ ∈ exist semi-open sets U in the topological field K containing λ and V in E containing x such that U.V W. ⊆ Since U V is semi-open in K E containing (λ, x) and ψ(U V) = U.V W, it follows that ψ is almost × × × ⊆ semi-continuous at (λ, x) and hence ψ is almost semi-continuous.

5. Conclusion

Topological vector spaces are a fundamental notion and play an important role in various advanced branches of mathematics like fixed point theory, operator theory, etc. This paper expounds the almost s-topological vector spaces which are basically a generalization of topological vector spaces. Section 1 and 2 make us familiar with the origin of almost s-topological vector spaces and the elementary concepts that are used to develop the theory of almost s-topological vector spaces. The third section presents some examples and counterexamples of almost s-topological vector spaces and interprets their relationships with some well-known existing spaces. Finally, some basic features of almost s-topological vector spaces are presented.

6. Acknowledgement

The authors are grateful to the referee for his valuable comments and suggestions. The first author is supported by UGC-India under the scheme of -JRF fellowship.

References

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