Ratio Mathematica Volume 40, 2021, pp. 113-121

On Characterization of δ-Topological
Vector Space

Shallu Sharma *
Tsering Landol†

Sahil Billawria‡

Abstract

The main objective of this paper is to present the study of δ-topological
vector space. δ-topological vector space is defined by using δ-open
sets and δ-continuous mapping which was introduced by J.H.H. Bayati[3]
in 2019. In this paper, along with basic inherent properties of the
space, δ-closure and δ-interior operators are discussed in detail. We
characterize some important properties like translation, dilation of the
δ-topological vector space and an example of δ-topological vector
space is also established.
Keywords: Regular open set, δ-open set, δ-closed set, δ-continuous
mapping and δ-topological vector space.
2020 AMS subject classifications:57N17, 57N99, 54A05. 1

*Department of Mathematics, University of Jammu JK-180006, India; shallu-
jamwal09@gmail.com.

†Department of Mathematics, University of Jammu, JK-180006, India; tseringlan-
dol09@gmail.com.

‡Department of Mathematics, University of Jammu, JK-180006, India;
sahilbillawria2@gmail.com.
1Received on January 16th, 2021. Accepted on June 23th, 2021. Published on June 30th, 2021.
doi: 10.23755/rm.v40i1.569. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper
is published under the CC-BY licence agreement.

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S. Sharma, T. Landol, S. Billawria

1 Introduction
In functional analysis, topological vector space is one of the fundamental

space being investigated by mathematicians due to the significant role played
by it in other branches of mathematics such as operator theory, fixed point the-
ory, variational inequality etc. The formalism of topological vector space be-
longs to Kolmogroff [4] who was the first to introduce a well structured notion
of topological vector space in his pioneering work done in 1934. Since then, it
has evolved further and many mathematicians have developed different general-
izations of topological vector space. In 2015, the notion of s-topological vector
space is being developed by M. Khan et.al.[5], which is one of the generaliza-
tion of topological vector space. Later, many other significant generalizations of
topological vector space are being introduced such as irresolute topological vec-
tor space [2], β-topological vector space [11], strongly preirresolute topological
vector space [9], almost s-topological vector spaces [10], etc.

2 Preliminaries
In this paper, (X,τ)(or simply X) always means topological space on which

no separation axioms are assumed unless stated explicitly. For a subset D of a
space X , we denote closure and interior by Cl(D) and Int(D) respectively and
neighborhood and δ-neighborhood of an element x in any topological space X is
denoted by N(x) and Nδ(x).

Definition 2.1. Let B be a subset of a topological space (X,τ). Then B is said to
be
(a) Regular open [12] if B = Int(Cl(B))
(b) Pre-open [6] if B ⊆ Int(Cl(B))
(c) β-open [1] if B ⊆ Cl(Int(Cl(B))).

Definition 2.2. A subset C of a topological space X is called
(a) Regular closed if X \C is open i.e. C = Cl(Int(C))
(b) Pre-closed if Cl(Int(C)) ⊆ C
(e) β-closed if Int(Cl(Int(C))) ⊆ C.

Definition 2.3. A subset D of a topological space X is said to be δ-open [13] if
for each x ∈ D, there exist a regular open set R such that x ∈ R ⊆ D.

Remark 2.1. Every regular open set is open and every open set is pre-open, while
the converse need not be true.

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On Characterization of δ-Topological Vector Space

Example 2.1. Let R be a set of real numbers with usual topology. Then Int(Cl(Z)) =
∅, which implies Z is not regular in topological space (R,τu). Also, set of ratio-
nal number denoted by Q is pre-open but neither regular open nor open set in
topological space (R,τu).

The complement of δ-open set is δ-closed. The concept of δ-closure and δ-
interior are introduced by Velicko [13] in 1968. The intersection of all δ-closed
sets in X containing a subset D ⊆ X is called δ-closure of D and is denoted by
Clδ(D). A point x ∈ Clδ(D) if and only if D ∩R 6= ∅, for a regular open set R
in X containing x. A subset C of X is δ-closed if and only if C = Clδ(C). The
union of all δ-open sets in X that are contained in D ⊆ X is called δ-interior of D
and is denoted by Intδ(D). A point x ∈ X is called δ-interior of D ⊆ X if there
exist a δ-open set U in X such that x ∈ U ⊆ D.

Definition 2.4. [4] Let X be a vector space over the field F(R or C). Let τ be a
topology on X such that
1) for each x,y ∈ X, and for each open neighborhood W of x+y in X, there exist
open neighborhoods U and V of x and y respectively in X such that U + V ⊆ W
2) for each λ ∈ F, x ∈ X and for each open neighborhood W of λ ·x in X, there
exist open neighborhoods U of λ in F and V of x in X such that U ·V ⊆ W .
Then, the (X(F),τ) is called topological vector space.

3 δ-Topological Vector Space
In this section, we give an examples of δ-topological vector space and further

illustrate the properties of this space.

Definition 3.1. [3] Let X be a vector space over field F(R or C) with standard
topology. Let τ be a topology over X such that the following conditions hold:
(a) For each x,y ∈ X and each open set R containing x + y, there exist δ-open
set P and Q containing x and y respectively, such that P + Q ⊆ R;
(b) For each λ ∈ F, x ∈ X and each open set R containing λx ∈ X, there exist
δ-open set P and Q containing λ and x respectively such that P.Q ⊆ R.
Then the pair (X(F),τ) is said to be δ-topological vector space.

Following are some examples of δ-topological vector.

Example 3.1. Let K = R with usual topology. Let X = R with base B =
{(a,b) : a,b ∈ R}. We shall show that (X(K),τ) is δ-topological vector space.
For, we will check the following:
(i) Let x,y ∈ X. Consider open set R = (x + y − �,x + y + �) in X containing

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S. Sharma, T. Landol, S. Billawria

x + y. Then we can choose δ-open sets P = (x−η,x+ η) and Q = (y−η,y + η)
in X containing x and y respectively such that P + Q ⊆ R, for each η < �

2
. This

establish the first condition of the definition of δ-topological vector space.
(ii) Let λ ∈ R and x ∈ X. Consider an open set R = (λx − �,λx + �) in X
containing λx. Then, we have the following cases:
Case I: If λ > 0 and x > 0, then we can choose δ-open set P = (λ−η,λ + η) in
R containing λ and Q = (x−η,x + η) in X containing x such that P.Q ⊆ R, for
each η < �

λ+x+1
.

Case II: If λ < 0 and x < 0, then λx > 0. We choose δ-open set P = (λ−η,λ+η)
of λ in R and Q = (x−η,x + η) of x in R such that P.Q ⊆ R, for η ≤ −�

λ+x−1 .
Case III: If λ > 0 and x = 0, (λ = 0 and x > O). We can choose δ-open
sets as P = (λ − η,λ + η) (resp.(−η,η)) containing λ in K and Q = (η,η)
(resp.(x − η,x + η)) containing x in R such that P.Q ⊆ R, for each η <
�

λ+1
(resp.(η < �

x+1
)).

Case IV: If λ = 0 and x < 0, (λ = 0 and x > 0). We can choose δ-open
sets as P = (η,η)(resp.(λ − η,λ + η)) containing λ in K and Q = (x −
η,x + η)(resp.(−η,η)) containing x in R, we have P.Q ⊆ R, for every η <
�

1−x(resp.(η <
�

1−λ)).
Case V: If λ = 0 and x = 0. Then, for δ-open set P = (η,η) and Q = (η,η) of λ
and x respectively such that P.Q ⊆ R, for each η <

√
�.

This proves that the pair (X(K),τ) is δ-TVS.

Example 3.2. Consider a vector space X = R of real number over the field K
with the topology τ = {φ,Qc,R}, where Qc denotes the set of irrational num-
bers and the field K is endowed with standard topology. Then (X(K),τ) is not
δ-topological vector space. For x,y ∈ Qc and open neighborhood Qc of x + y
in X, there doesn’t exist any δ-open sets P and Q containing x and y respectively
such that P + Q ⊆ Qc.

Theorem 3.1. [3] Let D be any open subset of δ-topological vector space X. Then
(a) x + D ∈ δO(X), for each x ∈ X;
(b) λD ∈ δO(X), for each non-zero scalar λ.

Theorem 3.2. Let C be any closed subset of a δ-topological vector space X, then
(a) x + C ∈ δC(X), for each x ∈ X;
(b) λC ∈ δC(X), for each non-zero scalar λ.

Proof: (a) Let y ∈ Clδ(x + C), z = −x + y and R be an open set in X con-
taining z. Then there exist δ-open set P and Q containing −x and y respectively,
such that P + Q ⊆ R. Also, y ∈ Clδ(x + C), y ∈ Q and Q is δ-open implies
there exist regular open set Q′ such that y ∈ Q′ ⊆ Q. So (x + C) ∩Q′ 6= ∅. Let
a ∈ (x + C) ∩Q′ ⇒ −x + a ∈ C ∩ (P ′ + Q′) ⊆ C ∩ (P + Q) ⊆ C ∩R 6= ∅.

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On Characterization of δ-Topological Vector Space

Hence z ∈ Cl(C) = C ⇒ −x + y ∈ C ⇒ y ∈ x + C. Thus, x + C ∈ δC(X),
for each x ∈ X.

(b) Assume that x ∈ Clδ(λC) and R be open neighborhood of y = 1λx ∈ X.
Since X is δ−TV S, there exist δ-open neighborhood P of 1

λ
in F and Q of x in X

such that P.Q ⊆ R. By hypothesis, (λC)∩Q′ 6= ∅, for regular open set Q′ subset
of Q containing x. Let a ∈ (λC)∩Q′. Now 1

λ
a ∈ C∩P ′.Q′ ⊆ C∩P.Q ⊆ C∩R

⇒ C ∩R 6= ∅ i.e. y is a limit point of C and so y = 1
λ
x ∈ Cl(C) = C, since C

is closed subset of X. Hence y ∈ λC. Since the inclusion λC ⊆ Clδ(λC) holds
generally, so Clδ(λC) = λC. Therefore, λC is δ-closed set in X. This completes
the proof.

Theorem 3.3. For any subset D of δ-topological vector space X,
(a) Clδ(x + D) ⊆ x + Cl(D), for each x ∈ X.
(b) x + Clδ(D) ⊆ Cl(x + D), for each x ∈ X.

Proof: (a) Let y ∈ Clδ(x + D) and consider z = −x + y in X. Let R be open
neighborhood of z. By hypothesis, there exist δ-open set P and Q containing -x
and y respectively such that P + Q ⊆ R. Existence of δ-open set confirms the
existence of regular open set P ′ and Q′ such that −x ∈ P ′ ⊆ P and y ∈ Q′ ⊆ Q.
Since y ∈ Clδ(x + D), (x + D) ∩ Q′ 6= ∅. Let a ∈ (x + D) ∩ Q′. Now,
−x+a ∈ D∩(P ′ +Q′) ⊆ D∩(P +Q) ⊆ D∩R 6= ∅, which implies z ∈ Cl(D).
Hence y ∈ x + Cl(D). This completes the proof.

(b) Let z ∈ x + Clδ(D). Then z = x + y, for some y ∈ Clδ(D). Let R
be any open neighborhood of z in X, then there exist δ-open neighborhood P and
Q of x and y respectively such that P + Q ⊆ R. Also, D ∩ Q′ 6= ∅, for regu-
lar open set Q′ ⊆ Q containing y which implies D ∩ Q 6= ∅. Let a ∈ D ∩ Q.
Then x + a ∈ (x + D) ∩ (P + Q) ⊆ (x + D) ∩ R 6= ∅, which implies z is a
limit point of x+D i.e z ∈ Cl(x+D). Hence the inclusion holds for each x ∈ X.

Theorem 3.4. For a subset D of δ-topological vector space X, the following are
valid:
(a) x + Int(D) ⊆ Intδ(x + D), for each x ∈ X.
(b) Int(x + D) ⊆ x + Intδ(D), for each x ∈ X.

Proof: (a) Assume y ∈ x + Int(D). Then,−x + y ∈ Int(D). Since
X is δ-topological vector space, there exist δ-open sets P containing -x and Q
containing y in X such that Q ⊆ Int(D). Also, δ-openness of P and Q im-
plies the existence of regular open set P ′ and Q′ such that −x ∈ P ′ ⊆ P
and y ∈ Q′ ⊆ Q satisfying P ′ + Q′ ⊆ P + Q ⊆ Int(D). In particular,

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S. Sharma, T. Landol, S. Billawria

−x + Q′ ⊆ Int(D) ⊆ D ⇒ Q′ ⊆ x + D. Thus there exist regular open set
Q′ containing y such that y ∈ Q′ ⊆ x + D, which implies y is δ-interior point of
x + D i.e. y ∈ Intδ(x + D). Hence the proof.

(b) Let y ∈ Int(x + D), then y = x + a, for some a ∈ D. By definition of
δ-topological vector space, there exist δ-open set P and Q such that x ∈ P,a ∈ Q
satisfying P +Q ⊆ Int(x+D). Hence, x+a ∈ P ′ +Q′ ⊆ P +Q ⊆ Int(x+D),
for each regular open set P ′ and Q′ such that x ∈ P ′ ⊆ P and a ∈ Q′ ⊆ Q. Now
x + Q′ ⊆ x + Q ⊆ Int(x + D) ⊆ x + D, which implies y ∈ x + Intδ(D). Hence
the inclusion Int(x + D) ⊆ x + Intδ(D) holds.

Theorem 3.5. Let D be any subset of δ-topological vector space X. Then the
following holds:
(a) λClδ(D) ⊆ Cl(λD), for every non-zero scalar λ.
(b) Clδ(λD) ⊆ λCl(D), for every non-zero scalar λ.

Proof The proof is trivial, omitted.

Theorem 3.6. Let X be a δ-topological vector space and D be any subset of X.
Then the following holds:
(a) Int(λD) ⊆ λIntδ(D), for every non-zero scalar λ.
(b) λInt(D) ⊆ Intδ(λD), for every non-zero scalar λ.

Proof The proof is trivial, omitted.

Theorem 3.7. Let C and D be any subset of a δ-topological vector space X. Then
Clδ(C) + Clδ(D) ⊆ Cl(C + D).

Proof: Let z ∈ Clδ(C) + Clδ(D). Then z = x + y, where x ∈ Clδ(C)
and y ∈ Clδ(D). Let R be an open neighborhood of z in X. By definition of
δ-topological vector space, there exist δ-open neighborhood P and Q of x and y
respectively such that P + Q ⊆ R. Since x ∈ Clδ(C), C ∩ P ′ 6= ∅ for regular
open set P ′ such that x ∈ P ′ ⊆ P and also y ∈ Clδ(D), D ∩Q′ 6= ∅ for regular
open set Q′ satisfying y ∈ Q′ ⊆ Q.
Let a ∈ C ∩ P ′ and b ∈ D ∩ Q′ ⇒ (a + b) ∈ (C + D) ∩ (P ′ + Q′) ⊆
(C + D) ∩ (P + Q) ⊆ (C + D) ∩ R ⇒ (C + D) ∩ R 6= ∅. Thus z is a
closure point of (C + D) i.e. z ∈ Cl(C + D). Hence the inclusion holds.

Theorem 3.8. For any subsets C and D of δ-topological vector space X. Then
C + Int(D) ⊆ Intδ(C + D).

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On Characterization of δ-Topological Vector Space

Proof: Let z ∈ C + Int(D) be arbitrary. Then z = x + y, for some
x ∈ C,y ∈ Int(D), which results in −x + z ∈ Int(D). By definition of δ-TVS,
there exist δ-open neighborhood P and Q containing -x and z respectively such
that P + Q ⊆ Int(D). Hence, there exist regular open sets P ′ and Q′ containing
-x and z respectively satisfying P ′ ⊆ P , Q′ ⊆ Q and P ′+Q′ ⊆ P +Q ⊆ Int(D).
In particular, −x + Q′ ⊆ Int(D) ⇒ Q′ ⊆ x + Int(D) ⊆ C + D. Hence, there
exist regular open set Q′ containing z such that z ∈ Q′ ⊆ C + D. Therefore, z is
δ-interior point of A + B. Hence the proof.

Definition 3.2. [8] A function f : X → Y is called δ-continuous if for each
x ∈ X and each open neighborhood Q of f(x), there exist open neighborhood P
of x such that f(Int(Cl(P)) ⊆ Int(Cl(Q)).
Lemma 3.1. [8] For a function f : X → X, the following are equivalent:
(a) f is δ-continuous.
(b) For each x ∈ X and each regular open set V containing f(x), there exist a
regular open set U containing x such that f(U) ⊆ V .
(c) f([A]δ) ⊂ [f(A)]δ, for every A ⊂ X.
(d) [f−1(B)]δ ⊂ f−1([B]δ), for every B ⊂ X.
(e) For every regular closed set F of Y, f−1(F) is δ-closed in X.
(f) For every δ-closed set V of Y, f−1(V ) is δ-closed in X.
(g) For every δ-open set V of Y, f−1(V ) is δ-open in X.
(h) For every regular-open set V of Y, f−1(V ) is δ-open in X.

Theorem 3.9. [3] Let X be δ-topological vector space, then the following are
true:
(a) the translation mapping ga : X → X defined by ga(b) = a + b, ∀b ∈ X is
δ-continuous.
(b) the mapping gλ : X → X defined by gλ(a) = λa, ∀a ∈ X is δ-continuous,
where λ is a fixed scalar.

Theorem 3.10. For a δ-topological vector space X, the mapping Φ : X×X → X
defined by Φ(x,y) = x + y, ∀x ∈ X ×X is δ-continuous.

Proof: Take arbitrary elements x, y in X and let R be regular open neighbor-
hood of x + y which implies R is open neighborhood of x+y. Then by hypoth-
esis, there exist δ-open neighborhood P and Q of x and y respectively such that
P + Q ⊆ R.
Also, by definition of δ-open set, there exist regular open neighborhood P ′ and
Q′ such that x ∈ P ′ ⊆ P and y ∈ Q′ ⊆ Q. This implies that Φ(P ′ × Q′) =
P ′ + Q′ ⊆ P + Q ⊆ R. Since P ×Q is regular open in X ×X(with respect to
product topology), it follows that Φ is δ-continuous.

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S. Sharma, T. Landol, S. Billawria

Theorem 3.11. For δ-topological vector space X, the mapping Ψ : K ×X → X
defined by Ψ(λ,x) = λx, ∀(λ,x) ∈ K×X is δ-continuous.

Proof: Let λ ∈ K and x ∈ X and R be a regular open neighborhood of λx in
X. Then there exist δ-open neighborhood P of λ in K and δ-open neighborhood
Q of x in X such that P.Q ⊆ R. Also, P ′.Q′ ⊆ P.Q ⊆ R, for regular open set
P ′ and Q′ contained in P and Q containing λ and x respectively. Since P ×Q is
regular in K × X, Ψ(P ′.Q′) ⊆ R. Hence, it follows that Ψ is δ-continuous for
arbitrary element λ ∈ K and x ∈ X.

Theorem 3.12. Let X be δ-topological vector space and Y be topological vector
space over the same field K. Let f : D1 → D2 be a linear map such that f is
continuous at 0. Then f is δ-continuous.

Proof: Let 0 6= x ∈ X and V be regular open set and hence open in Y con-
taining f(x). Since translation of an open set is open in topological vector space,
which implies V − f(x) is open in Y containing 0. Since f is continuous at 0,
there exist open set U in X containing 0 such that f(U) ⊆ V − f(x). Also, by
linearity of f implies that f(x + U) ⊆ V . By theorem 3.1, x + U is δ-open and
hence there exist regular open set Q such that Q ⊆ x + U. Hence, f(Q) ⊆ V .

4 Conclusions
δ-Topological vector space is an extension of topological vector space and

this paper give an insight into this space. We presented the space with new exam-
ples and inherent properties. Moreover, important characterization of the space is
studied in this paper.

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