On Almost Pretopological Vector Spaces

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

DOI: 10.4236/oalib.1104937 Nov. 28, 2018 1 Open Access Library Journal

On Almost Pretopological Vector Spaces

Shallu Sharma, Madhu Ram, Sahil Billawria

Department of Mathematics, University of Jammu, Jammu, India

Abstract

In this paper, we introduce the notion of almost pretopological vector spaces and present some examples of almost pretopological vector spaces. Almost pretopological vector spaces are defined by using regular open sets and pre-open sets. The relationships of almost pretopological vector spaces with certain other types of spaces are provided. Along with some useful results, it is proved that in almost pretopological vector spaces, translation and scalar multiple of a regular open (resp. regular closed) set are pre-open (resp. pre-closed).

Subject Areas

Functional Analysis, Topology

Keywords

Pre-Open Sets, Regular Open Sets, δ-Open Sets, Almost Pretopological Vector Spaces

1. Introduction

The concept of topological vector spaces was first introduced and studied by Kolmogroff [1] in 1934. Due to nice properties, the notion earns a great impor-tance in fixed point theory, operator theory and various other advanced branches of Mathematics. In 2015, M. Khan et al. [2] introduced the notion of s-topological vector spaces which is a generalization of topological vector spaces. Later, in 2016, M. Khan and M.A. Iqbal [3] introduced and studied another class of spaces called irresolute topological vector spaces which is contained in the class of s-topological vector spaces but independent of topological vector spaces. Ibrahim [4] initiated the study of α-topological vector spaces.

The purpose of the present paper is to introduce a new class of spaces, namely almost pretopological vector spaces. These spaces are defined by using regular

How to cite this paper: Sharma, S., Ram, M. and Billawria, S. (2018) On Almost Pretopological Vector Spaces. Open Access Library Journal, 5: e4937.

https://doi.org/10.4236/oalib.1104937

Received: September 22, 2018 Accepted: November 25, 2018 Published: November 28, 2018

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

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DOI: 10.4236/oalib.1104937 2 Open Access Library Journal

open sets and pre-open sets. The relationships of almost pretopological vector spaces with certain other types of spaces are presented. Some basic properties of almost pretopological vector spaces are given.

2. Preliminaries

Throughout this paper,

(

X,τ

)

(or simply X) means a topological space. For a subset A X⊆ _{, the closure of}_A_{and the interior of}_A_{are denoted by}Cl A

( )

and Int A

( )

respectively. The notation  (resp. ) represents the set of real numbers (resp. the set of complex numbers) and  (resp. η) represents negligi-bly small positive number.

Definition 2.1. Let X be a topological space. A subset A of X is called 1) regular open if A Int Cl A=

(

( )

)

_.

2) pre-open [5] if A Int Cl A⊆

(

( )

)

_.

3) semi-open [6] if A Cl Int A⊆

(

( )

)

_.

Definition 2.2. A subset A of a topological space X is said to be δ-open [7] if for each ∈_A_{, there exists a regular open set}_U_in_X_{such that} x U∈ ⊆A_.

The complement of a regular open (resp. pre-open, semi-open, δ-open) set is called regular closed (resp. pre-closed, semi-closed, δ-closed [7]). The intersec-tion of all pre-closed (resp. δ-closed) sets containing a subset A of X is called the pre-closure (resp. δ-closure) of A and is denoted by pCl A

( )

(resp. Cl Aδ

( )

. It

is known that a subset A of X is pre-closed (resp. δ-closed) if and only if

( )

A pCl A= _(resp. A Cl A= δ

( )

). A point x pCl A∈

( )

(resp. Cl Aδ

( )

) if and

only if A U ≠ ∅ for each pre-open (resp. regularly open) set U in X contain-ing x. The union of all pre-open (resp. δ-open) sets in X that are contained in

A X⊆ _{is called the pre-interior (resp.}_δ_{-interior) of}_A_{and is denoted by}

( )

pInt A (resp. Int Aδ

( )

). A point x X∈ is called a pre-interior point of

A X⊆ _{if there exists a pre-open set}_U_in_X_{such that} x U∈ ⊆A_{. The set of all}

pre-interior points of A is equal to pInt A

( )

. It is well-known that a subset

A X⊆ _{is pre-open (resp.}_δ_{-open) if and only if}_{A pInt A}=

( )

(resp.

( )

A=Intδ A ).

The family of all regular open (resp. pre-open, pre-closed) sets in X is denoted by RO(X) (resp. PO(X), PC(X)). The family of all pre-open sets in X containing

x is denoted by PO(X, x). If A PO X B PO Y∈

( )

, ∈

( )

(X and Y are topological spaces), then A B PO X Y× ∈

(

×

)

(with respect to product topology).

Definition 2.3. [8] A function f X: →Y _{from a topological space}_X_{to a}

topological space Y is called almost precontinous if for each x X∈ and each regular open set V Y⊆ _containing f x

( )

, there exists a pre-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 vector space over the field F ( or ). Let T be a topology on L such that

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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 neighbourhood 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 T( )F,

)

is called topological vector space.

Definition 2.5. [2] Let L be a vector space over the field F ( or ) and Let

T 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

λ

x, there ex-ist semi-open sets U in F containing

λ

_and_V_in_L_containing_x_{such that}

U V W⋅ ⊆ .

Then the pair

(

L T( )F,

)

is called s-topological vector space.

Definition 2.6. [3] Let L be a vector space over the field F ( or ) and Let

T be a topology on L such that

3) 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

4) For each λ∈F x L, ∈ and each semi-open set W in L containing

λ

x, 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 T( )F,

)

is called irresolute topological vector space.

3. Almost Pretopological Vector Spaces

In this section, we define almost pretopological vector spaces and investigate their relationships with certain other types of spaces. Some general properties of almost pretopological vector spaces are also discussed.

Definition 3.1. Let E be a vector space over the field K, where K=_ or 

with standard topology. Let

τ

be a topology on E such that the following con-ditions are satisfied:

1) For each x y E, ∈ _{and each regular open set}W ⊆E_containing x y+ _,

there exist pre-open sets U and V in E containing x and y respectively, such that

U V W+ ⊆ _{, and}

2) For each x y E, ∈ and each regular open set W⊆E_containing

λ

_x, there exist pre-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 pretopological vector space

(writ-ten in short, APTVS).

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DOI: 10.4236/oalib.1104937 4 Open Access Library Journal Example 3.1. Consider the field K= with standard topology. Let E=

be endowed with the topology

τ

on E generated by the base B=

{

( )

a b a b, : , ∈_

}

_.

We show that

(

E( )K ,

τ

)

is an almost pretopological vector space. For this, we

have to check the following:

1) Let x y X, ∈ _{. Consider any regular open set}W =

(

x y+ −,x y+ +

)

in E

containing x y+ . Then we can choose pre-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

<₂ _{. This verifies the first condition of the definition of almost}

pretopological vector spaces.

2) Let

λ

∈ and x E∈ . Consider a regular open set W=

(

λx−,λx+

)

in E containing

λ

x. We have following cases:

Case (I). If

λ

>0 and x>0, then for the choice of pre-open sets

(

,

)

U= λ η λ η− + in  containing

λ

and V=

(

x−η,x+η

)

in E contain-ing x, we have U V W⋅ ⊆ _{for each} .

1 x η λ < + + 

Case (II). If

λ

<0 and x<0, then

λ

x>0. We can choose pre-open neighborhoods U=

(

λ η λ η− , +

)

_of

λ

_in _and V =

(

x−η,x+η

)

_of_x_in

E such that U V W⋅ ⊆ _{for sufficiently appropriate}

1 x η λ − ≤ + −  _.

Case (III). If

λ

=0 and x>0, (resp.

λ

>0 and x=0). Then, for the se-lection of pre-open neighborhoods U= −

(

η η,

)

(resp. U=

(

λ η λ η− , +

)

) of

λ

in  and V =

(

x−η,x+η

)

_(resp. V= −

(

η η,

)

_{) of}_x_in_E_{, we see that}

U V W⋅ ⊆ _{for each}

1

x

η< +

 _(resp.

1 η λ < +  _).

Case (IV). If

λ

=0_and x<0_{, (resp.}

λ

<0_and x=0_{). Then, for the}

se-lection of pre-open sets U= −

(

η η,

)

_(resp.U=

(

λ η λ η− , +

)

_{) in}_

contain-ing

λ

_and V =

₍

x−η,x+η

₎

_(resp.V = −

(

η η,

)

_{) in}_E_containing_x_{, we have}

U V W⋅ ⊆ _{for every}

1 x

η< −

 _(resp. 1 η λ < −  _).

Case (V). If

λ

=0 and x=0 . Then, for pre-open neighborhoods

(

,

)

U= −η η _{) of}

λ

_in_ℝ_and U= −

(

η η,

)

_of_x_in_E_{, we have}U V W⋅ ⊆ _for

each η< .

This proves that the pair

(

E( )R,

τ

)

is an almost pretopological vector space.

After tasting this example, an immediate question that comes into mind: is there any other topology on  which turns it out an almost pretopological vector space. The answer is in affirmative. In fact, there are many other topolo-gies on  which turn it out an almost pretopological vector space. Let us present some of them.

Example 3.2. Consider the field K= with standard topology. Let E=

be the vector space over the field K, where E is endowed with the topology

{ }

{

, 0 ,

}

τ= ∅ _ . Then

(

E_{( )}_K ,

τ

)

is an almost pretopological vector space.

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denotes the set of irrational numbers}. Then

(

E( )R,

τ

)

is an almost

pretopolog-ical vector space.

So far we have presented examples of almost pretopological vector spaces, we now present an example which lies beyond the class of almost pretopological vector spaces.

Example 3.4. Consider the field K= with standard topology. Let E=

with the topology

τ

generated by the base =

{

( )

a b, , 1, : , ,

[

c a b c

)

∈_

}

. Then

( )

(

E_R,

τ

)

_{is not an almost pretopological vector space because [1, 2) is a regular}

open set in E containing 1 = 1.1 but there do not exist pre-open sets U in K con-taining 1 and V in E containing 1 such that U V⋅ ⊆

[

1, 2

)

_.

Remark 3.1. Clearly, by definition, every topological vector space is almost pretopological vector space but the converse is not true in general because, Ex-ample 3.2 and ExEx-ample 3.3 are almost pretopological vector spaces which are not topological vector spaces.

Remark 3.2. Almost pretopological vector spaces are independent of s-topological vector spaces. It is easy to check that Example 3.2 is not an s-topological vector space. Furthermore, Example 3.4 is not an almost pretopo-logical vector space which is shown an s-topopretopo-logical vector spaces in [2].

Theorem 3.1. Let A be any δ-open subset of an almost pretoplogical vector space E. then the following are true:

(i) x A PO E+ ∈

( )

_{for each} x E∈

(ii) λA PO E∈

( )

_{for each non-zero scalar}

λ

_.

Proof. (i) Let y x A∈ + _{. Then} –x y A+ ∈ _{. Since}_A_is_δ_{-open set in}_E_{, there}

exists W RO E∈

( )

such that –x y W+ ∈ ⊆A. Since E is an APTVS, there exist pre-open sets U in E containing –x and V in E containing y such that

U V W+ ⊆ _{and therein,}y V∈ ⊆ +x A_{. Consequently,}y pInt x A∈

₍

+

₎

_and

therefore pInt x A

(

+

)

= +x A_{. Thus,} x A PO E+ ∈

( )

_.

(ii) Let x∈

λ

A. Since A is δ-open, there exists a regularly open set W in E

such that 1x W A

λ ∈ ⊆ . By definition of almost pretoplological vector spaces,

there exist pre-open sets U PO K∈ _ ,_λ1_

  , V PO E x∈

(

,

)

such that

U V W⋅ ⊆ ⊆A_{. This gives that}V ⊆λA_{. Consequently,} λA PO E∈

( )

_.

Corollary 3.1.1. Let A and B be any δ-open subsets of an almost pretopologi-cal vector spaces E. Then A B PO E+ ∈

( )

_.

Corollary 3.1.2. Let A be any δ-open subset of an almost pretopological vec-tor space E. Then the following are true:

(i) x A Int Cl x A+ ⊆

(

+

)

_{for each} x E∈ .

(ii) λA PC E∈

( )

for each non-zero scalar

λ

.

Theorem 3.2. Let F be any δ-closed subset of an almost pretopological vector space E. Then the following are valid:

(i) x F PC E+ ∈

( )

_{for each} x E∈ _.

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Proof. (i) Suppose that y pCl x F∈

(

+

)

_{. If we consider} z= − +x y_{and let}_W

be any open set in E containing z, then Int Cl W

(

( )

)

is regularly open set in E

such that z W∈ ⊆Int Cl W

(

( )

)

_{. The definition of almost pretopological vector}

spaces yields pre-open sets U and V in E containing –x and y respectively, such that U V+ ⊆Int Cl W

(

( )

)

. Since y pCl x F∈

(

+

)

, there is r V∈ 

(

x F+

)

. Now –x r F+ ∈ 

(

U V+

)

⊆F Int Cl W

(

( )

)

⇒F Int Cl W

(

( )

)

≠ ∅ . There-fore, z∈Cl Fδ

( )

=F and, as a result, y x F∈ + . This proves that

(

)

x F pCl x F+ = + and hence x F PC E+ ∈

( )

.

(ii) Suppose that x pCl F∈

( )

λ _{. Let}_W_{be any open set in}_E_containing 1x

λ .

By definition of almost pretopological vector spaces, there exist pre-open sets U

in K containing _λ1 and V in E containing x such that U V⋅ ⊆Int Cl W

(

( )

)

_.

Since x pCl F∈

( )

λ , V

( )

λF ≠ ∅. So, there is r E∈ such that r V∈ and

r∈

λ

F _{. Now,}1r F

₍

U V

₎

F Int Cl W

(

_{( )}

)

F Int Cl W

(

_{( )}

)

λ ∈  ⋅ ⊆  ⇒  ≠ ∅ .

Therefore, x∈λCl Fδ

( )

=λF and thereby, pCl F

( )

λ =λF. Hence the

asser-tion follows.

Corollary 3.2.1. Let F be any δ-closed subset of an almost pretopological vec-tor space E. Then the following are valid:

(i) Cl Int x F

(

+

)

⊆ +x F _{for each} x E∈ _.

(ii) Cl Int F

(

( )

λ

)

⊆λF_{for each non-zero scalar}

λ

_.

Now, we investigate further properties of almost pretopological vector spaces by using their basic idea.

Theorem 3.3. For any subset A of an almost pretopological vector space E,

( )

(

)

x pCl A+ ⊆Cl x Aδ + for each x E∈ .

Proof. Let y pCl A∈

( )

_{and let}_W_{be any open neighborhood of} z x y= + _in E. Then, there exist pre-open sets U PO E x∈

(

,

)

_andV PO E y∈

(

,

)

_{such that}

( )

(

)

U V+ ⊆Int Cl W _{. Since} y pCl A∈

_{( )}

_{, there is} a A V∈ _ _{. Now}

(

) (

)

(

( )

)

x a+ ∈ +x A _ U V+ ⊆ x A+ _Int Cl W _{implies that}z Cl x A∈ δ

₍

+

₎

;

that is, x pCl A+

( )

⊆Cl x Aδ

(

+

)

. This completes the proof.

If the set A in Theorem 3.3 is replaced by the δ-open set, then the following inclusion holds in almost pretopological vector spaces.

Theorem 3.4. For any δ-open subset of an almost pretopological vector space E, x pCl A+

( )

⊆Cl x A

(

+

)

_{for each} x E∈ .

Proof. Suppose that y pCl A∈

( )

_{and let}_W_{be any open neighborhood of}

z x y= + _in_E_{. Since}Int Cl W

(

( )

)

is regularly open, by definition of almost pretopological vector spaces, we get pre-open sets U and V in E containing x and

y respectively, such that U V+ ⊆Int Cl W

(

( )

)

_{. Since} y pCl A∈

_{( )}

_, A V_ ≠ ∅_.

So, there is a A V∈  .

Consequently, x a+ ∈ +

(

x A

) (

_ U V+

) (

⊆ x A+

)

_Int Cl W

(

( )

)

(

)

(

( )

)

(

)

( )

x A Int Cl W x A Cl W

⇒ + ≠ ∅



 .

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⇒Int Cl x A

(

+

)

W ≠ ∅ ⇒Cl x A W

(

+

)

 ≠ ∅.

Since W is open,

(

x A+

)

W≠ ∅. This results in z Cl x A∈

(

+

)

; that is,

( )

(

)

x pCl A+ ⊆Cl x A+ _{. The proof is finished.}

The analogs of Theorem 3.3 and 3.4 respectively, are the following results: Theorem 3.5. For any subset A of an almost pretopological vector space E,

( )

pCl A Clδ A

λ⋅ ⊆ λ _{for each non-zero scalar}

λ

.

Proof. The proof follows along the lines of the proof of Theorem 3.3.

Theorem 3.6. For any δ-open subset A of an almost pretopological vector space E, λ⋅pCl A

( )

⊆Cl

( )

λA _{for each non-zero scalar}

λ

_.

Proof. Follows along the lines of the proof of Theorem 3.4.

Theorem 3.7. Let A be any subset of an almost pretopological vector space E. Then the following assertions are true:

(i) pCl x A

(

+

)

⊆ +x Cl Aδ

( )

for each x E∈ .

(ii) pCl A

( )

λ ⊆λClδ

( )

A for each non-zero scalar

λ

.

Proof. (i) Let y pCl x A∈

(

+

)

_{. Consider} z= − +x y_{and let}_W_{be any open}

set in E containing z. Then, there exist pre-open sets U V PO E, ∈

( )

such that

x U

− ∈ _, y V∈ _andU V+ ⊆Int Cl W

(

( )

)

_{. Now,} y pCl x A∈

₍

+

₎

⇒

(

x A+

)

V ≠ ∅ ⇒_{there is} r∈

₍

x A+

₎

V. This gives,

(

)

(

( )

)

x r A U V A Int Cl W

− + ∈ _ + ⊆ _ _{. This proves that}z Cl A∈ δ

_{( )}

; that is,

( )

y x Cl A∈ + δ . Hence the assertion follows.

(ii) Let x pCl A∈

( )

λ _{. Consider} y 1x

λ

= . Let W be any open set in E

con-taining y. By definition of almost pretopological vector space, we get pre-open sets U PO K∈ _ ,1_λ_

  and V PO E x∈

(

,

)

such that U V⋅ ⊆Int Cl W

(

( )

)

. Since

( )

x pCl A∈ λ ,

( )

λA V ≠ ∅. So, there is a∈

( )

λA V. Now

(

)

(

( )

)

1_{a A U V} _{A Int Cl W}

λ ∈  ⋅ ⊆  showing that A Int Cl W

(

( )

)

≠ ∅.

Therefore, y Cl A∈ δ

( )

; that is, x∈λCl Aδ

( )

. Thereby the assertion follows.

Theorem 3.8. For any open subset A of an almost pretopological vector space E, the following are true:

(i) pCl x A

(

+

)

⊆ +x Cl A

( )

(ii) pCl A

( )

λ ⊆λCl A

( )

λ

.

Proof. (i) Assume that y pCl x A∈

(

+

)

_{. Let} z= − +x y_and_W_{be any open}

neighborhood of z. Then, there exist pre-open sets U containing –x and V con-taining y in E such that U V+ ⊆Int Cl W

(

( )

)

_{. Since}_V_{is pre-open,}

(

x A+

)

V ≠ ∅. So, there is a E∈ such that a x A∈ + and a V∈ . This im-plies − + ∈x a A U V_

(

+

)

⊆A Int Cl W_

(

( )

)

_⇒A Int Cl W_

(

( )

)

≠ ∅_{. Since}_A

is open, A W_ ≠ ∅_{. This results in} y x Cl A∈ +

_{( )}

. Thus,

(

)

( )

pCl x A+ ⊆ +x Cl A _.

(ii) Assume that x pCl A∈

( )

λ _{. Let}_W_{be any open set in}_E_containing 1x

λ .

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DOI: 10.4236/oalib.1104937 8 Open Access Library Journal 1

λ and V in E containing x such that U V⋅ ⊆Int Cl W

(

( )

)

. Since V is pre-open,

there is a∈

( )

λA V . Thereby we get, _λ1a∈_λ1.

(

)

(

( )

)

A U V A Int Cl W

λ _ ⋅ ⊆ _ _⇒ A Int Cl W_

(

( )

)

≠ ∅ _⇒ A Cl W_

_{( )}

≠ ∅_.

Since A is open, A W_ ≠ ∅_{. This shows that} 1x Cl A

_{( )}

λ ∈ and hence

( )

x∈λCl A _{. Thereby the assertion follows.}

Theorem 3.9. For any subset A of an almost pretopological vector space E, the following assertions are valid:

(i) Int x Aδ

(

+

)

⊆ +x pInt A

( )

for each x E∈ .

(ii) x Int A+ δ

( )

⊆pInt x A

(

+

)

for each x E∈ .

Proof. (i) Let y Int x A∈ δ

(

+

)

. Then y x a= + for some a A∈ . Since

(

)

Int x Aδ + is δ-open, there exists W RO E∈

( )

such that

(

)

y W∈ ⊆Int x Aδ + . By definition of almost pretopological vector spaces, there

exist pre-open sets U V PO X, ∈

( )

such that x U a V∈ , ∈ _and

(

)

U V W+ ⊆ ⊆Int x Aδ + ⊆ +x A ⇒ y x a x V U V= + ∈ + ⊆ + ⊆ +x A ⇒

( )

y x a x pInt A= + ∈ + . Hence the assertion follows.

(ii) Let y∈Intδ

( )

A . Then there exists U RO E∈

( )

such that

( )

y U∈ ⊆Intδ A . This gives that x y x U+ ∈ + ⊆ +x A. In view of Theorem 3.1,

( )

x U PO E+ ∈ _{and consequently,} x y pInt x A+ ∈

(

+

)

_{. Thus,}

( )

(

)

x Int A+ δ ⊆ pInt x A+ .

The analog of Theorem 3.9 is the following:

Theorem 3.10. For any subset Aof an almost pretopological vector space E, the following are true:

(i) Intδ

( )

λA ⊆λpInt A

( )

λ

.

(ii) λInt Aδ

( )

⊆pInt

(

λ⋅A

)

λ

.

Theorem 3.11. Let A be any δ-open subset of an almost pretopological vector space E. Then Int x A

(

+

)

⊆ +x Int Aδ

( )

for each x E∈ .

Proof. Let y Int x A∈

(

+

)

. Then –x y A+ ∈ _{. Since}_A_is_δ_{-open, there exists}

( )

W RO E∈ _{such that}− + ∈x y W⊆ =A I t Anδ

( )

⇒ y x I∈ + ntδ

( )

A . This

proves that Int x A

(

+

)

⊆ +x nt AI δ

( )

.

Theorem 3.12. Let A be any semi-closed set in an almost pretopological vec-tor space E. Then

(i) x Int A+

( )

⊆pInt x A

(

+

)

(ii) λInt A

( )

⊆pInt

(

λ⋅A

)

λ

. Proof. (i) Assume that y x Int A∈ +

( )

_{. Since}_A_{is semi-closed,}

( )

Int A ∈RO E _{and as a result, there exist pre-open sets}_U_and_V_in_E_such

that − ∈x U y V, ∈ _and U V+ ⊆Int A

( )

_{. Now,} − +x V U V⊆ + ⊆Int A

( )

⊆A

⇒ y V∈ ⊆ +x A_{. Since}_V_{is pre-open,} y pInt x A∈

(

+

)

_{and hence}

( )

(

)

x Int+ A ⊆pIn x At + .

(9)

regu-DOI: 10.4236/oalib.1104937 9 Open Access Library Journal

larly open, there exist pre-open sets U in the topological field K containing _λ1 and V in E containing x such that U V⋅ ⊆Int A

( )

_{. Whence we get} x V∈ ⊆λA

and hence y pInt A∈

( )

λ _{. Hence the assertion follows.}

Presenting properties of some special functions on almost pretopological vec-tor spaces.

Theorem 3.13. For an almost pretopological vector space E, the following are always true:

(1) the translation mapping T Ex: →E defined by T yx

( )

= + ∀x y x y E, , ∈ ,

is almost precontinuous.

(2) the multiplication mapping M Eλ: →E defined by M xλ

( )

=λx , x E∈ _{, (λ is non-zero fixed scalar), is almost precontinous.}

Proof. (1) Let y E∈ _{be an arbitrary. Let}_W_{be any regular open set in}_E

containing T yx

( )

. Then, by the definition of almost pretopological vector

space, there exist pre-open sets U in E containing x and V in E containing y

such that U V W+ ⊆ _{. This results in} x V W+ ⊆ ⇒ _{T V}_x

_{( )}

⊆_W. This indi-cates that Tx is almost precontinuous at y and hence Tx is almost

preconti-nuous.

(2) Let x E∈ _and_W_{be any regular open set in}_E_containing

λ

x. Then, there exist pre-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 Vλ ⊆W showing that Mλ is almost precontinous at x. since x E∈ was

an arbitrary, it follows that Mλ is almost precontinous.

Theorem 3.14. For an almost pretopological vector space E, the mapping :E E E

φ × → _{defined by} φ

₍

x y,

₎

= + ∀x y,

₍

x y,

₎

∈ ×E E_,_{is almost}

preconti-nuous.

Proof. Let

(

x y,

)

∈ ×E E_{and let}_W_{be any regular open set in}_E_{such that}

(

x y,

)

W

φ ∈ . Since E is an APTVS, there exist pre-open sets U and V in E such that x U y V∈ , ∈ _andU V W+ ⊆ _{. Since} U V× _{is pre-open in}E E× (with respect to the product topology) such that

(

x y,

)

∈ ×U V and

(

U V

)

U V W

φ × = + ⊆ _{, it follows that}φ_{is almost precontinuous at}

₍

x y,

₎

and consequently, φ is almost precontinuous.

Theorem 3.15. For an almost pretopological vector space E, the mapping :K E E

ψ × → _{defined by}ψ λ

(

,x

)

=λx,∀

(

λ,x

)

∈ ×K E, is almost preconti-nuous.

Proof. Let λ∈K x E, ∈ _{. Let}_W_{be any regular open set in}_E_containing

λ

x_.

Then, there exist pre-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 pre-open in}K E×

containing

(

λ,x

)

_and ψ

(

U V×

)

= ⋅ ⊆U V W _{, it follows that}

ψ

_{is almost}

precontinuous at

(

λ,x

)

_{and hence}

ψ

_{is almost precontinuous.}

Acknowledgements

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The second and the third authors are supported by UGC-India under the scheme of NET-JRF fellowship.

Conflicts of Interest

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

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