Topological Vector Spaces Derived From Topological Hypervector

Spaces

Abstract: We introduce topological hypervector spaces on a topological ﬁeld, in the sense of Tallini, and study

some basic properties of this hyperspaces. In this regards we study the relationship between the topology on

a hypervector spaces and its complete part. In particular we show that if every open subset of a topological

hypervector space is a complete part then its fundamental vector space induced is a topological vector space.

Finally, we study the quotient space of topological hypervector spaces and the derived topological space of a

topological hypervector space with respect its fundamental relegation.

Key–Words: Topological hypervector space, Fundamental relation, Complete part, Topological fundamental vector

space, Homeomorphism.

Received: October 9, 2022. Revised: August 14, 2023. Accepted: September 19, 2023. Published: October 6, 2023.

1 Introduction

The concept of hypergroups as a generalization

of groups was ﬁrstly introduced in 1934 at the

8thCongress of Scandinavian Mathematicians by F.

Marty [25]. In the following, it has been studied

and extended by many researchers. Indeed the no-

tion of hyperstructures is a generalization of classi-

cal algebraic structures. M.S. Tallini introduced the

notion of hypervector spaces over a ﬁeld, and stud-

ied the basic properties of this hyperspaces( for more

details see [31, 32]. Also, some important proper-

ties of (fuzzy) hypervector spaces were studied in

[3, 4, 5, 6, 7, 8, 9, 15].

On of the main topic in theory of hyperstruc-

tures is the study of regular and strongly regular re-

lations. In particular, the fundamental relations on a

hyperstructure, as the smallest strongly relation on the

hyperstructure such that its derived quotient spaces

with respect this relation become an algebraic struc-

ture, for example for special hyperstructures such as

semihypergroups, hypergroups, hyperrings, hypervec-

tor spaces and etc., their corresponding derived al-

gebraic structures with the fundamental relations is

semigroup, group, ring, module, vectors space and

etc. The fundamental relations play important role in

the study algebraic hyperstructures. In fact, these re-

lation construct a connection between the categories

of hyperstructures and categories of algebraic struc-

tures(for more details see [2]). The fundamental re-

lation β∗on hypergroups introduced by Koskas[24],

and was studied mainly by Corsini [11] and Vougiouk-

lis [33].

Later on, Freni [16] introduced the γ∗-relation on

a hypergroup, as a generalization of the β∗-relation.

Then B. Davvaz et al. [1], R. Ameri et al. [9] and

M. Hamidi et al. [18] introduced the ν∗-relation,

ξ∗-relation and τ∗-relation, respectively. In [33] T.

Vougiouklis introduced the fundamental relation ε∗

of Hv-vector space (a general class of hypervector

spaces) and in [4], R. Ameri et al. deﬁned the fun-

damental relation ε∗for a given hypervector space V,

over a classical ﬁeld K(in the sense of Tallini) as the

smallest equivalence relation on Vsuch that V/ε∗is a

classical vector space over K.

The notion of topological(transposition) hyper-

groups introduced and studied by R. Ameri([3]), in

this paper notions of a (pseudo, strong pseudo) topo-

logical (transposition) hypergroups and introduced

and the relationships between pseudo topological

polygroups and topological polygroups investigated.

Also, Heidari et al.( [19]) studied the concept of topo-

logical hypergroups as a generalization of topological

groups. Since then, many researchers have worked

on topological hyperstructures ( for more details see

[10],[12],[14],[17],[19],[20],[21], [22],[26],[27],[28],

[29],[30]).

One of classes of topological hyperstructures is

hypervector spaces. The notion of a topological hy-

pervector space introduced in [34]. In this paper, we

follow [34] and study more properties of topological

1REZA AMERI, 2M. HAMIDI, 2A. SAMADIFAM

1Department of Mathematics, School of Mathematics, Statistics and Computer Sciences,

University of Tehran, Tehran, IRAN

2Department of Mathematics, University of Payame Noor, Tehran, IRAN

PROOF

DOI: 10.37394/232020.2023.3.7

Reza Ameri, M. Hamidi, A. Samadifam

E-ISSN: 2732-9941

Volume 3, 2023

hyperstructures. In this regards, we consider various

kinds of topologies on power set of a topological hy-

pervector space and use them to introduce the various

kinds of topological hypervector spaces. In particu-

lar, we consider the upper topology over P∗(V), the

family of all nonempty subsets of V, and prove that

if (V, +,◦, K, T)is a topological hypervector space

such that every its open subset is a complete part, then

the quotient space V/ε∗, the set of all equivalence

classes by ε∗, is a topological vector space. Finally,

we consider a topological vector space (V, +,·, K, T)

and its subhyperspace W, to form the topological hy-

pervector space (V , +,◦, K, T)and prove that two

hyperspaces V /ε∗and V/W are homeomorphic.

2 Preliminaries

In this section, we review some deﬁnitions and results

which we need to development our paper.

Atopological group is a group (G, .)which is

also a topological space such that the multiplication

map (g, h)→gh from G×Gto G, and the inverse

map g→g−1from Gto G, are both continuous.

Similarly, a topological ring or a topological ﬁeld

are deﬁned. A topological vector space is a vector

space Xover a topological ﬁeld K(most often

the real or complex numbers with their standard

topologies) that is endowed with a topology such

that vector addition + : X×X→Xand scalar

multiplication.:K×X→Xare both continuous

functions with respect to product topologies on

X×X, and on K×Xand X, respectively, such

that the mapping x7→ −x= (−1)x, is continuous

and the topology on Xis compatible with its additive

group structure.

Let Gbe a nonempty set and P∗(G)be the fam-

ily of all nonempty subsets of G. A mapping ·:

G×G−→ P∗(G), where is called a hyperopera-

tions, or a hypercomposition on G, that is for all x,

yof G,·(x, y), denoted by x◦y, or simply by xy is

a nonempty subset of G, and it called hyperproduct

of xand y. An algebraic system (G, ·1,·2,...,·n)is

called a hyperstructure, the pair (G, ·)endowed with

only hyperoperation is called a hypergroupoid. For

every two nonempty subsets Aand Bof Gby A·B

we means Sa∈A,b∈Ba·b.

Deﬁnition 1 ([31]) Let Kbe a ﬁeld and (V, +) be

an abelian group. A hypervector space over Kis a

quadruple (V, +,◦, K), where ◦is a mapping ◦:K×

V→P∗(V), such that for all a, b ∈Kand x, y ∈V

the following conditions hold:

(H1)a◦(x+y)⊆a◦x+a◦y;

(H2) (a+b)◦x⊆a◦x+b◦x;

(H3)a◦(b◦x)=(ab)◦x;

(H4)a◦(−x)=(−a)◦x=−(a◦x);

(H5)x∈1◦x,

where for all A, B ∈P∗(V), A +B={a+b|a∈

A, b ∈B}.

Remark 2 If in (H1)the equality holds, then the hy-

pervector space is called strongly right distributive. If

in (H2)the equality holds, the hypervector space is

called strongly left distributive. A hypervector space

is called strongly distributive hypervector space, if it

is both strongly left and strongly right distributive.

Clearly, every classical vector space over a ﬁeld

Kis also an strongly distributive hypervector space

over K, with the operations on Vand K, which is

called a trivial hypervector space. A nonempty subset

Wof Vis called a subhyperspace, if Wis itself a hy-

pervector space with the external hyperoperation on

V, i.e. for all a∈Kand x, y ∈W, x −y∈Wand

a◦x⊆W. Let Ω = 0 ×0V, where 0Vis the zero

of (V, +). If Vis either strongly right distributive,

or left distributive, then Ωis a subgroup of (V, +).

An strongly right distributive hypervector space is

strongly left distributive.

Lemma 3 Let Xand Ybe topological spaces and

let f:X→Y. Then the following statements are

equivalent:

(1) fis continuous;

(2) for all open subset Uof Y, f−1(U)is open in X;

(3) for all x∈Xand all open subset Vof Xcon-

taining f(x), there exists an open subset Uof X

containing xsuch that f(U)⊆V.

Lemma 4 [3] Let (X, T)be a topological space,

then the family Bconsisting of all

SU={W∈P∗(X) : W⊆U, U ∈ T },

is a base for a topology on P∗(X). This topology is

denoted by T∗.

Lemma 5 [3,13]Let (X, ◦)be a hypergroupoid and

Tbe a topology on X. Then the following assertions

are equivalent:

(1) for any U∈ T , the set {(x, y)∈X×X:x◦y⊆

U}is open in X×X;

PROOF

DOI: 10.37394/232020.2023.3.7

Reza Ameri, M. Hamidi, A. Samadifam

E-ISSN: 2732-9941

Volume 3, 2023

(2) for every x, y ∈Xand U∈ T such that x◦

y⊆U, there exist Ux, Uy∈ T containing x, y

respectively, such that Ux◦Uy⊆U;

(3) for every x, y ∈Xand U∈ T such that x◦

y⊆U, there exist Ux, Uy∈ T containing x, y

respectively,such that a◦b⊆Ufor any a∈Ux

and b∈Uy.

Let (V, +,◦, K)be hypervector space over a topolog-

ical ﬁeld Kand Tbe a topology on V. In the follow-

ing we use the topology T∗on P∗(V)and the product

topology on V×V.

3 Topological Hypervector Spaces

In this section we introduce the concept of topological

hypervector spaces and study some their properties.

Deﬁnition 6 Let (V, +,◦, K)be a hypervector space

over a topological ﬁled Kand (V, T)be a topological

space. Then (V, +,◦, K, T)is said to be a topological

hypervector space (thvs)

if the operations + : V×V→V, (x, y)7→ x+y,

i:V→V, x 7→ −xand the hyperoperation

◦:K×V→P∗(V),(a, x)7→ a◦xare continuous.

Example 7 Every topological vector space

(V, +,·, K, T)with hyperoperation a◦x={a·x}is

a topological hypervector space over K.

Example 8 Every hypervector space (V, +,◦, K)

with trivial topology Tis a topological hypervector

space. Since, if we have T={∅, V }then

T∗={∅, SV}={∅, P ∗(V)}.

Example 9 Let K=V=Z2={0,1}. Then

(V, +,◦, K)is a hypervector space, where a◦x=

{0,1}for any a∈Kand x∈V. Let T=

{∅,{0},{1}, V }be a topology on V=K. We have

T∗={∅,{{0}},{{1}},{{0},{1}}, P ∗(V)}. It is

clear that Vis a topological hypervector space.

Example 10 By considering the external hyperoper-

ation ◦:R×R2→P∗(R2), a◦(x, y) = a·x×Rthen

(R2,+,◦, R)is a strongly distributive hypervector

space. The family B={(x, y) : a < x < b, y ∈R}

is a base for a topology on R. Then (R2,+,◦, R, T)

is a topological hypervector space.

Example 11 Let:

◦:R×R→P∗(R), a ◦x={a·x, −a·x}

be a external hyperoperation on R.Then (R, +,◦, R)

is a hypervector space, but it is neither the right dis-

tributive nor the left distributive. With standard topol-

ogy on R, (R, +,◦, R, T)is a topological hypervec-

tor space.

Example 12 Let:

◦:R×R→P∗(R), a ◦x={a·x, −a·x, 0}

be a external hyperoperation on R. Then (R, +,◦, R)

is a hypervector space, but it is neither the right dis-

tributive nor the left distributive. With standard topol-

ogy on V=Rand discrete topology on K=R,Vis

a topological hypervector space.

Topological hypervector spaces are a generalization of

topological vector spaces but some characteristics of

topological vector spaces are not valid in topological

hypervector spaces. If Vis a thvs, (V, +) is a topo-

logical group.

Lemma 13 Let Vbe a thvs. Then

(1) for ﬁxed x∈V, the map y7→ x+yis a homeo-

morphism of Vonto V;

(2) if Uis open and x∈V, then x+Uis open; if U

is open and Ais any subset of V, then A+Uis

open;

(3) for ﬁxed a∈K, the map x7→ a◦xis continuous,

but not necessarily open. In Example 11, U=

(2,3) is open and 2◦(2,3) = (−6,−4) ∪(4,6)

is also open, but in the Example 12, U= (2,3)

is open and 2◦(2,3) = (−6,−4) ∪ {0} ∪ (4,6)

is not open in R.

The complete parts were introduced for the ﬁrst time

by Koskas [24]. Then, this concept was studied by

many authors. Let (V, +,◦, K)be a hypervector

space over Kand Abe a nonempty subset of V. We

say that Ais a complete part of V, if for nonzero nat-

ural number n, for all a1, . . . , anof K, and for all

x1, . . . , xnof V, the following implication holds:

A∩

i=1

ai◦xi6=∅=⇒

i=1

ai◦xi⊆A.

Theorem 14 Let Vbe a thvs, A ⊆Vand Ube an

open subset of V, such that Uis a complete part of V.

Then A⊆a−1◦Uif and only if a◦A⊆Ufor all

a∈K.

Suppose that A⊆a−1◦Uand x∈A. So x∈a−1◦U,

and there exists u∈U, such that x∈a−1◦uthus,

a◦x⊆a◦(a−1◦u) = 1◦u. We have u∈1◦u, u ∈U,

which implies that 1◦u⊆Usince Uis complete part.

Therefore a◦x⊆U.

Conversely, suppose that a◦A⊆Uand a∈K.

Then, we have A⊆a−1◦(a◦A)⊆a−1◦U.

Theorem 15 Let Ube an open subset of a thvs, such

that Uis a complete part. Then

PROOF

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Reza Ameri, M. Hamidi, A. Samadifam

E-ISSN: 2732-9941

Volume 3, 2023

(1) a◦Uis an open subset of Vfor any a∈K, a 6=

(2) for any subset Aof Kand for 06=a∈A, A ◦U

is open.

(1) The map Pa:V→P∗(V), PA:x7→ a◦xis

continuous. For a6= 0 we have

P−1

a−1(SU) = {x∈V:a−1◦x⊆U}=a◦U,

thus a◦Uis open. (2) Since the union of open subsets

is open, therefore A◦U=Sa∈Aa◦Uis open.

4 Topological Fundamental Vector

Spaces

In this section, the concept of a topological fundamen-

tal vector space derived of a topological hypervector

space is introduced. Let (V, +,◦, K)be a hypervec-

tor space over K. The fundamental relation ε∗of V

was introduced by T. Vougiouklis in [33] as the small-

est equivalence relation on Hv−vector space, a gen-

eral class of hypervector spaces, such that the quo-

tient V/ε∗is a vector space over K. In the following,

we introduce the fundamental relation on hypervector

spaces in the sense of Tallini, and study the relation-

ship between Vand V/ε∗in the way of [4].

let Ube the set of all ﬁnite linear combinations of

elements of Vwith coefﬁcient in K, that follows

U=(n

i=1

ai◦xi:ai∈K, xi∈V, n ∈N).

Now, consider the ε-relation over Vby

xεy ⇐⇒ ∃u∈U:{x, y} ⊆ u, ∀x, y ∈V.

Let ε∗be the transitive closure of ε. We deﬁne addi-

tion operation and scalar multiplication on V/ε∗by

{ ⊕ :V/ε∗×V /ε∗→V /ε∗ε∗(x)⊕ε∗(y) = {ε∗(t) : t∈ε∗(x)+ε∗(y)}

and

{  :K×V /ε∗→V/ε∗aε∗(x) = {ε∗(z) : z∈a◦ε∗(x)}.

Theorem 16 (see [33])Let (V, +,◦, K)be a hyper-

vector space over K. Then,

(1) ε∗(a◦x) = ε∗(y)for all y∈a◦x, ∀a∈K, ∀x∈

V, where ε∗(a◦x) = Sb∈a◦xε∗(b).

(2) ε∗(x)⊕ε∗(y) = ε∗(x+y).

(3) ε∗(0) is the identity element of (V/ε∗,⊕).

(4) (V/ε∗,⊕,, K)is a vector space over K.

The vector space (V/ε∗,⊕,, K)is called funda-

mental vector space of V.

Theorem 17 Let (V, +,◦, K)be a hypervector space

over Kand (V/ε∗,⊕,, K)be the fundamental vec-

tor space of V. Then the canonical map π:V→

V/ε∗, such that π(x) = ε∗(x), is an epimorphism.

Let x, y ∈Vand a∈K, we see that π(x+y) =

π(x)⊕π(y). Now we show that π(a◦x) = aπ(x).

We have π(a◦x) = ε∗(a◦x) = ε∗(y)for all y∈a◦x.

On the other hand, we have y∈a◦x, x ∈ε∗(x)that

implies y∈a◦ε∗(x)thus, aπ(x) = aε∗(x) =

{ε∗(z) : z∈a◦ε∗(x)}=ε∗(y).

Let Xbe a topological space and ∼be any equiv-

alence relation on X. The quotient set of all equiva-

lence classes is given by the X/ ∼={[x] : x∈X}.

We have the canonical map or quotient map π:X→

X/ ∼, x 7→ [x],and we deﬁne a topology on X/ ∼

by setting that: U⊆X/ ∼is open if and only if

π−1(U)is open in X. Then it is easy to verify that:

(1) the canonical map πis continuous.

(2) the quotient topology on X/ ∼is the ﬁnest topol-

ogy on X/ ∼s.t. πis continuous.

(3) the canonical map πis not necessarily open or

closed.

Theorem 18 Let (V, +,◦, K)be a topological hy-

pervector space over K, such that every open sub-

set of Vis a complete part. Then the canonical map

π:V→V/ε∗is open.

Let Wbe an open subset of Vand x∈π−1(π(W)),

we have π(x)∈π(W)thus there exists v∈Wsuch

that π(x) = π(v)and x∈ε∗(v).Hence, there ex-

ist a1, . . . , an∈Kand x1, . . . , xn∈V, such that

{x, v} ⊆ Pn

i=1 ai◦xi. Since Wis open so there ex-

ists an open subset Uof V, such that v∈U⊆W.

Hence we have v∈U∩Pn

i=1 ai◦xiand Uis com-

plete part, so x∈Pn

i=1 ai◦xi⊆U⊆W. Thus,

x∈U⊆π−1(π(W)). Therefore, π(W)is open in

V/ε∗.

Theorem 19 Let (V, +,◦, K, T)be a topological hy-

pervector space over Ksuch that every open subset

of Vis a complete part . Then, (V/ε∗,⊕,,T∗)is

a topological vector space over K, where T∗is the

quotient topology on V/ε∗.

By Theorem16, (V/ε∗,⊕,)is a vector space. We

show that the mappings

⊕: (π(x), π(y)(7→ π(x)⊕π(y)

PROOF

DOI: 10.37394/232020.2023.3.7

Reza Ameri, M. Hamidi, A. Samadifam

E-ISSN: 2732-9941

Volume 3, 2023

and

: (a, π(x)) 7→ aπ(x)

are continuous, where ⊕=⊕ε∗and =ε∗.

(1) Let Ube an open subset of V/ε∗and x, y ∈U,

such that π(x)⊕π(y)∈U. So, we have

π(x+y)∈Uor x+y∈π−1(U). Since

π−1(U)is open in Vand Vis thvs, it follows

that there exist open subsets U1, U2of Vsuch

that x∈U1, y ∈U2and U1+U2⊆π−1(U)or

π(U+U2)⊆U, thus π(U1)⊕π(U2)⊆U.

(2) Let Ube an open subset in V/ε∗and a∈K, x ∈

Vsuch that aπ(x)∈U. There exists z∈

a◦π(x)and we have π(z)∈Uso z∈π−1(U).

Since a◦x⊆a◦π(x), so a◦x⊆π−1(U). Thus

there exists open subsets U1and U2containing

aand xfrom Kand V, respectively such that

U1◦U2⊆π−1(U)hence U1π(U2)⊆U.

Since we have π(U1◦U2) = π(Sa∈U1a◦U2) =

Sa∈U1π(a◦U2)

=Sa∈U1(aπ(U2)) = U1π(U2).

Atopological vector space (tvs) is a vector space

Vover a topological ﬁeld Kequipped with a topology

such that the maps (x, y)7→ x+yand (a, x)7→ a·x

are continuous from X×X→Xand K×X→X,

respectively.

Let X, Y be two vector space over K. A mapping

f:X→Yis called homomorphism if for any x, y ∈

Xand for any λ∈K,

f(x+y) = f(x) + f(y), f(λx) = λf(x).

A bijective homomorphism between two vector

spaces Xand Yover Kis called algebraic isomor-

phism and we say that Xand Yare algebraically iso-

morphic X∼

=Y. Let Xand Ybe two tvs on K. A

topological isomorphism (homeomorphism)from X

to Yis a algebraic isomorphism which is also contin-

uous and open.

Let Vbe a tvs and W⊆Vbe a linear subspace

of V. The quotient space V/W consists of cosets x+

W= [x]and the quotient map π:V→V/W is

deﬁned by π(x) = x+W.

We construct a topological hypervector space

such as Vusing a classical topological vector space

Vand its linear subspace Wand prove that V /ε∗and

V/W are homeomorphic.

Theorem 20 For a linear subspace Wof a tvs V ,

the quotient map π:V→V/W is a continuous and

open map, when V /W is equipped with the quotient

topology.

The mapping πis continuous by the deﬁnition of the

quotient topology. Let Ube open in V. Then we have

π−1(π(U)) = U+W=[

v∈W

(U+v),

since U+vis open for any v∈W, hence

π−1(π(U)) is open in Vas a union of open sets.

Therefore, π(U)is open in V/W .

Theorem 21 [23] Let Wbe a linear subspace of a

tvs V . Then the quotient space V/W equipped with

the quotient topology is a tvs.

Let (V, +,·, K)be a classical vector space and

Wbe a linear subspace of Vand let V=V. Then

(V , +,◦, K)is a strongly distributive hypervector

space where

◦:K×V→P∗(V), a ◦x=a·x+W,

Vis said to be the associated hypervector space con-

cerning the vector space V.

Theorem 22 Let (V, +,·, K)be a classical vector

space and Wbe a linear subspace of V.

Then V /ε∗∼

=V/W.

We deﬁne a mapping f:V /ε∗→V /W by

f(ε∗(x)) = x+W.

(1) the mapping fis well-deﬁned. Let ε∗(x) =

ε∗(y), it follows that xε∗yand we have x∈

1◦x=x+W, y ∈1◦y=y+W, since

the two sets x+Wand y+Ware equal or dis-

joint subset of V/W , thus x+W=y+Wand

so f(ε∗(x)) = f(ε∗(y)).

(2) fis linear. Since, f(ε∗(x) + ε∗(y)) = f(ε∗(x+

y)) = x+y+W

=x+W+y+W

=f(ε∗(x)) + f(ε∗(y)).and f(aε∗(x)) =

f(ε∗(z)), z ∈a◦ε∗(x), on the other hand,

a·x∈1◦(a·x) = a◦x⊆a◦ε∗(x)which

implies that f(ε∗(z)) = f(ε∗(a·x)) = a·x+W

=a◦(x+W) = a◦f(ε∗(x)).

(3) The mapping fis surjective. For one-to-one

property of f, let ε∗(x)∈Ker(f). Then

f(ε∗(x)) = x+W=Wthus x∈W. There-

fore, ε∗(W) = ε∗(0) = 0V /ε∗, which implies

that fis one-to-one. Consequently, fis an alge-

braic isomorphism.

Theorem 23 Let (V, +,·, K, T)be a tvs. Then

(V , +,◦, K, T)is a topological hypervector space.

PROOF

DOI: 10.37394/232020.2023.3.7

Reza Ameri, M. Hamidi, A. Samadifam

E-ISSN: 2732-9941

Volume 3, 2023

It is enough to show that the mapping ◦:K×V→

P∗(V), a ◦x=a·x+Wis continuous. Let Ube

an open subset of V. the mapping “◦” is continuous if

and only if {(a, x)∈K×V:a◦x⊆U}is an open

subset of K×Vfor all U∈ T . We have a◦x⊆U⇒

a·x+W⊆U. Since a·x∈a·x+W⊆Uand

the mapping “·” is continuous, there exist U1and U2

containing aand xrespectively, such that U1·U2⊆U.

Theorem 24 Let (V, +,·, K, T)be a tvs and Wbe

a linear subspace of V. Then V /ε∗and V /W are

topologically isomorphic.

By Theorem 22, the map

f:V /ε∗→V/W, f(ε∗(x)) = x+W

is algebraic isomorphism. It is enough to show that

fis continuous and open. Suppose that Ais open in

V/W . We show that π−1(f−1(A)) is open in V/W .

Let x∈π−1(f−1(A)), then π(x)∈f−1(A), and so

f(π(x)) ∈A, thus x+W∈A. Since the canon-

ical map q:V→V/W is continuous, there ex-

ists an open subset Uxcontaining xof Vsuch that

Ux⊆q−1(A). We show that Ux⊆π−1(f−1(A)). If

t∈Ux, then t+W∈A, and so t∈π−1(f−1(A)).

Therefore, π−1(f−1(A)) is open in V, and fis con-

tinuous.

Now suppose that Ais an open subset of V /ε∗. We

show that f(A)is an open subset of V/W . Let

x+W∈f(A), then ε∗(x)∈A. Since the canonical

mapping π:V→V /ε∗is continuous, there exists

an open subset Uxcontaining xof Vsuch that Ux⊆

π−1(A). We show that {z+W:z∈Ux} ⊆ f(A). If

z∈Ux, then z+W=f(ε∗(x)) ∈f(A), thus f(A)

is open in V/W . Therefore fis open.

5 Conclusion

In this paper the notion of upper topology for topolog-

ical hypergroups in [3] has been generalized to topo-

logical hypervector spaces, for short THVS( in the

sense of Tallini) and the topological properties of them

was investigated. Also, by considering the fundamen-

tal relation ε∗, as the smallest equivalence relation on

a THVS space V, the topological behavior of the fun-

damental vector space V/ε∗was investigated. In par-

ticular, it was proved that if in a topological hyper-

vector space Vevery open sets is a complete part,

then the canonical map π:V→V/ε∗is a open

mapping and the fundamental vector space V/ε∗is

a topological vector space, too. Finally, for a topo-

logical vector space (V, +,·, K, T)and its subhyper-

space Wof V, it was shown that the quotient space

(V/W, +,◦, K, T)is also a topological hypervector

space and V/W/ε∗is homeomorphic to ∼

=V/W.

We hope that is paper encourage intrusted re-

searchers to work in this topic and develop more

results of topological hypervector spaces, in partic-

ular extend this results to other classes of hyper-

vector spaces, such as Krasner hypervector spaces,

HV−vector spaces and etc.

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DOI: 10.37394/232020.2023.3.7

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E-ISSN: 2732-9941

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