EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 915-922
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Note on (i, j)−mX − β−Exterior Sets in Biminimal
Structure Spaces

Torsak Prasertsang1, Patarawadee Prasertsang1,∗

1 Faculty of Science and Engineering, Kasetsart University, Chalermprakiat Sakon Nakhon
Province Campus, Sakon Nakhon 47000, Thailand

Abstract. In that paper, the concept of (i, j) −mX − β−exterior sets in a biminimal structure
space (BSS) and a biminimal structure subspace (BSs) were introduced. Based on properties of
BSS and BSs, some new notions and several properties of those sets dealing with this space were
obtained in both of BSS and BSs. Some examples were given to illustrate the effectiveness of these
results.

2020 Mathematics Subject Classifications: 22A05, 22A15

Key Words and Phrases: Exterior sets, Biminimal structure spaces, (i, j) −mX − β−exterior
sets,(i, j)−mX − β−closed sets,(i, j)−mX − β−open sets

1. Introduction

A general space in mathematics, topology space has been widely studied in every field
of mathematics as a fundamental concept including the definition of limits, continuity,
neighborhoods, closed sets, open set and connnectedness among others. In 2020, T. M.
Al-shami et. al. [1] focused their attention on topology space. The concept of supra semi
limit points of a set and new types of separation axioms using supra semi-open sets were
introduced to minimize the conditions of topology for other reasons. Some applications of
supra preopen sets on supra topological spaces was studied by M. E. El-Shafei and et. al
[5]. The concept of supra prehomeomorphism maps and the concepts of supra limit and
supra boundary points of a set with respect to supra preopen sets and their properties
were introduced. More recently, A. Mhemdi and T. M. Al-shami [6] defined the functional
separation axioms on general topology and provided some notions of them. The notions
of almost SD-compact and almost SD-Lindelöf spaces, nearly SD-compact and nearly SD-
Lindelöf spaces, and mildly SD-compact and mildly SD-Lindelöf spaces were investigated
by T. M. Al-shami and T. Noiri [13].

The above discussion motivated the current study, of some branches of topology. The

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4003

Email addresses: torsak.p@ku.th (T. Prasertsang), patarawadee.s@ku.th (P. Prasertsang)

http://www.ejpam.com 915 © 2021 EJPAM All rights reserved.



T. Prasertsang, P. Prasertsang / Eur. J. Pure Appl. Math, 14 (3) (2021), 915-922 916

concept of a biminimal structure space (BSS) and some properties of m1
Xm

2
X−closed

sets and m1
Xm

2
X−open sets in BSSs were introduced by Boonpok [2] in 2010. That

is, (X,m1
X ,m

2
X) is called a biminimal structure space, where X is a nonempty set and

m1
X ,m

2
X are minimal structures on X where minimal structures are defined by giving

P (X) as the power set of a nonempty set X. A subfamily mX of P (X) is called a minimal
structure on X if ∅ ∈ mX and X ∈ mX .

Biminimal structure space has been of wide interest in studying in Topology. Further-
more, Boonpok et al. [2–4] provided some properties of them to as a preliminary for the
current study, such as (i, j)−mX−α−closed, (i, j)−mX−α−open, (i, j)−mX−β−closed
and (i, j)−mX − β−open, which are advantageous for studying BSS. Later, S. Sompong
and S. Muangchan [10, 11] studied the notion of exterior sets in this space and obtained
some characterizations and fundamental properties of those sets. E. Subha and N. Na-
gaveni [12] studied strongly minimal generalized closed set in BSSs and obtained some
properties for the set. Later, P. Prasertsang and S. Sompong [8, 9] studied the concept
of (i, j) −mX − α−boundary and exterior sets and (i, j) −mX − β−boundary sets and
provided some fundamental properties of such sets dealing with those spaces as well, which
was relevant to the current research.

In this paper, the concepts of (i, j)−mX−β−exterior sets are introduced and some fun-
damental properties of those sets are obtained and some examples are given for completing
some properties. Lastly, the special properties of a biminimal structure subspace and the
product of those sets are defined and then some fundamental properties are provided.

2. Preliminaries

In this section we recall some notions, notations and previous results.

Definition 1. [11] Let P (X) be the power of nonempty set X. A subfamily mX of P (X)
is called a minimal structure (briefly m− structure) on X if ∅ ∈ mX and X ∈ mX .

Definition 2. [2] Let X be a nonempty set and m1
X ,m

2
X be minimal structures on X.

The triple (X,m1
X ,m

2
X) is called a biminimal structure space (briefly BSS) or a bispace

(briefly bi m−space [7])

Lemma 1. [3] Let (X,m1
X ,m

2
X) be a biminimal structure space and A be a subset of X.

It follows that:
1. A is (i, j)−mX−regular −closed if and only if A = mi

XCl(m
j
XInt(A)),

2. A is (i, j)−mX−semi−closed if and only if mi
XInt(m

j
XCl(A)) ⊆ A,

3. A is (i, j)−mX−preclosed if and only if mi
XCl(m

j
XInt(A)) ⊆ A,

4. A is (i, j)−mX − α−closed if and only if mi
XCl(m

j
XInt(m

i
XCl(A))) ⊆ A,

5. A is (i, j)−mX − β−closed if and only if mi
XInt(m

j
XCl(m

i
XInt(A))) ⊆ A.

Definition 3. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A be a subset of

X. Then, mij
X −β−closure of A and the mij

X −β−interior of A where i, j = 1, 2 and i 6= j.
are defined as follows:



T. Prasertsang, P. Prasertsang / Eur. J. Pure Appl. Math, 14 (3) (2021), 915-922 917

1. mij
XClB(A) =

⋂
{F : A ⊆ F, F is (i, j)−mX − β−closed},

2. mij
XIntB(A) =

⋃
{U : U ⊆ A,U is (i, j)−mX − β− open}.

Lemma 2. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A,B be subsets of

X, the following hold:
1. mij

XClB(∅) = ∅, mij
XClB(X) = X, mij

XIntB(∅) = ∅ and mij
XIntB(X) = X,

2. A ⊆ mij
XClB(A) and mij

XIntB(A) ⊆ A,
3. If A ⊆ B then mij

XClB(A) ⊆ mij
XClB(B) and mij

XIntB(A) ⊆ mij
XIntB(B).

Lemma 3. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A be a subset of X.

The following properties hold:
1. mij

XClB(A) is (i, j)−mX − β−closed,

2. mij
XIntB(A) is (i, j)−mX − β−open,

3. A is (i, j)−mX − β−closed if and only if mij
XClB(A) = A,

4. A is (i, j)−mX − β−open if and only if mij
XIntB(A) = A.

Lemma 4. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A,B be subsets of

X, the following hold:
1. If A and B are (i, j)−mX − β−closed then A ∩B is (i, j)−mX − β−closed,
2. If A and B are (i, j)−mX − β−open then A ∪B is (i, j)−mX − β−open.

Lemma 5. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A a subset of X:

1. mij
XIntB(X\A) = X\mij

XClB(A),

2. mij
XClB(X\A) = X\mij

XIntB(A).

Lemma 6. [9] Let (X,m1
X ,m

2
X) be a biminimal structure space and A a subset of X, for

any i, j = 1, 2 and i 6= j,
1. mij

XBdrB(A) ∩mij
XIntB(X\A) = ∅,

2. mij
XClB(X\A) = mij

XBdrB(A) ∪mij
XIntB(A),

3. X = mij
XIntB(A) ∪mij

XBdrB(A) ∪mij
XIntB(X\A) is a pairwise disjoint union.

Definition 4. [9] Let (X,m1
X ,m

2
X) be a biminimal structure spaces and W be a subset of

X. Define m1
W and m2

W as follows: m1
W = A ∩W : A ∈ m1

X and m2
Y = B ∩W : B ∈ m2

X .
A triple (W,m1

W ,m
2
W ) is called a biminimal structure subspace of (X,m1

X ,m
2
X).

Let (W,m1
W ,m

2
W ) be a biminimal structure subspace of (X,m1

X ,m
2
X), and A be a

subset of W. The (i, j)−mW − β−closure and (i, j)−mW − β−interior of A with respect
to mij

W are denoted by mij
WClB(A) and mij

W IntB(A), respectively (for i = 1, 2 and i 6= j).

Then, mij
WClB(A) = W ∩mij

XClB(A),

mij
W IntB(A) = W ∩mij

XIntB(A)

and mij
WBdrB(A) = W ∩mij

XBdrB(A)

consequently, mij
WExtB(A) = W ∩mij

XExtB(A).



T. Prasertsang, P. Prasertsang / Eur. J. Pure Appl. Math, 14 (3) (2021), 915-922 918

3. Main Results

In this section, we introduce the concepts of (i, j)−mX−β−exterior sets in biminimal
structure space which contains some characterizations and several fundamental properties
of those sets.

Definition 5. Let (X,m1
X ,m

2
X) be a biminimal structure space, A be a subset of X and

x ∈ X. Then, x is called (i, j)−mX − β−exterior point of A if x ∈ mij
XIntB(X\A). The

set of all (i, j)−mX−β−exterior point of A are denoted by: mij
XExtB(A) where i, j = 1, 2

and i 6= j.

By the Definition 5, mij
XExtB(A) = mij

XIntB(X\A) = X\mij
XClB(A).

Example 1. Let X = {1, 2, 3}. Define m−structures m1
X and m2

X on the biminimal
structure space X as follows:

m1
X = {∅, {2}, {1, 3}, X} and m2

X = {∅, {1}, {3}, {1, 2}, {2, 3}, X}.
We have that: m12

XExtB({1, 2}) = {3} and m21
XExtB({1, 2}) = ∅.

Lemma 7. Let (X,m1
X ,m

2
X) be a biminimal structure space, A be a subset of X. Then,

for any i, j = 1, 2 and i 6= j, the following statements hold:
1. mij

XExtB(∅) = X and mij
XExtB(X) = ∅,

2. mij
XExtB(A) ∩A = ∅ and mij

XExtB(A) ∩mij
XClB(A) = ∅,

3. mij
XExtB(A) ∩mij

XExtB(X\A) = ∅ and mij
XExtB(A) ∩mij

XBdrB(A) = ∅,
4. X = mij

XIntB(A) ∪mij
XBdrB(A) ∪mij

XExtB(A) is a pairwise disjoint union.

Proof. Assume that (X,m1
X ,m

2
X) is a biminimal structure space and A is a subset of

X.
1. Since mij

XClB(∅) = ∅ and mij
XClB(X) = X, we obtain: mij

XExtB(∅) = X\∅ = X

and mij
XExtB(X) = X\X = ∅.

2. By Lemma 2 (2), X\mij
XClB(A) ⊆ X\A, (X\mij

XClB(A)) ∩ A ⊆ ∅. That is:

mij
XExtB(A) ∩A = ∅. It follow that: mij

XExtB(A) ∩mij
XClB(A) = ∅.

3. It follows by Lemma 6 (1).
4. It is obvious by Definition 5 and Lemma 6 (2), (3).

Theorem 1. Let (X,m1
X ,m

2
X) be a biminimal structure space and A,B be subsets of X

with A ⊆ B. Then, for i, j = 1, 2 and i 6= j,
1. mij

XExtB(B) ⊆ mij
XExtB(A),

2. mij
XExtB(B) ⊆ X\mij

XBdrB(A).

Proof. Assume that (X,m1
X ,m

2
X) is a biminimal structure space and A,B are subsets

of X with A ⊆ B. For any i, j = 1, 2 and i 6= j,
1. From Lemma 2 (3), mij

XClB(A) ⊆ mij
XClB(B) yields

X\mij
XClB(B) ⊆ X\mij

XClB(A),

mij
XExtB(B) ⊆ mij

XExtB(A).



T. Prasertsang, P. Prasertsang / Eur. J. Pure Appl. Math, 14 (3) (2021), 915-922 919

2. By Lemma 7 (3), mij
XExtB(A) ⊆ X\mij

XBdrB(A) and by (1), we have

mij
XExtB(B) ⊆ X\mij

XBdrB(A).

Corollary 1. Let (X,m1
X ,m

2
X) be a biminimal structure space and A be subsets of X.

Then, for i, j = 1, 2 and i 6= j,
1. mij

XExtB(A) ⊆ mij
XExtB(mij

XIntB(A)),

2. mij
XExtB(mij

XClB(A)) ⊆ mij
XExtB(A),

3. mij
XExtB(A) ⊆ X\mij

XBdrB(mij
XIntB(A)),

4. mij
XExtB(mij

XClB(A)) ⊆ X\mij
XBdrB(A).

Proof. It follows by Theorem 1 and Lemma 2 (2).

Theorem 2. Let (X,m1
X ,m

2
X) be a biminimal structure space and A be a subset of X.

Then, for any i, j = 1, 2 and i 6= j, the following statement are true:
1. A is (i, j)−mX − β−closed if and only if mij

XExtB(A) = X\,
2. A is (i, j)−mX − β−open if and only if mij

XExtB(X\A) = A.

Proof. Assume that (X,m1
X ,m

2
X) is a biminimal structure space and A is a subset of

X.
1. (=⇒) Suppose thatA is (i, j)−mX−β−closed. Then, mij

XExtB(A) = X\mij
XClB(A) =

X\A.
(⇐=) Suppose that mij

XExtB(A) = X\A. It means that X\mij
XClB(A) = X\A.

Since A ⊆ mij
XClB(A), then mij

XClB(A) = A. Finally, A is (i, j)−mX − β−closed.
2. (=⇒) Suppose that A is (i, j)−mX−β−open. Then, X\A is (i, j)−mX−β−closed.

Using (1), mij
XExtB(X\A) = X\(X\A) = A.

(⇐=) Suppose that mij
XExtB(X\A) = A. We have A = X\mij

XClB(X\A)

= X\(X\mij
XIntB(A)) = mij

XIntB(A)). Hence, A is (i, j)−mX − β−open.

Corollary 2. Let (X,m1
X ,m

2
X) be a biminimal structure space and A be a subset of X.

Then, for i, j = 1, 2 and i 6= j,
1. mij

XExtB(mij
XClB(A)) = mij

XExtB(A),

2. mij
XExtB(X\mij

XExtB(A)) = mij
XExtB(A)

Proof. This follows by Theorem 2 immediately.

From Example 1, m12
XExtB({2})∪m12

XExtB({1, 3}) 6= m12
XExtB({2}∩{1, 3}), whereas

m12
XExtB({2})∪m12

XExtB({1, 2}) = m12
XExtB({2} ∩ {1, }). Therefore, it needs some con-

ditions to show that
mij

XExtB(A) ∪mij
XExtB(B) = mij

XExtB(A ∩B),
which found in the next result. Similarly, the following equation

mij
XExtB(A ∪B) = mij

XExtB(A) ∩mij
XExtB(B)

is true if it has some appropriate conditions.



T. Prasertsang, P. Prasertsang / Eur. J. Pure Appl. Math, 14 (3) (2021), 915-922 920

Theorem 3. Let (X,m1
X ,m

2
X) be a biminimal structure space, A,B be subsets of X.

Then, for any i, j = 1, 2 and i 6= j, we have:
1. If A and B are (i, j)−mX − β−closed, then

mij
XExtB(A) ∪mij

XExtB(B) = mij
XExtB(A ∩B).

2. If A,B and A ∪B are (i, j)−mX − β−closed, then
mij

XExtB(A ∪B) = mij
XExtB(A) ∩mij

XExtB(B).

Proof. Assume that (X,m1
X ,m

2
X) is a biminimal structure space and A,B are subsets

of X.
1. Assume that A and B are (i, j) − mX − β−closed. Therefore, A ∩ B is also

(i, j)−mX − β−closed. By Theorem 2 (1),
mij

XExtB(A ∩B) = X\A ∩B
= (X\A) ∪ (X\B)
= mij

XExtB(A) ∪mij
XExtB(B).

2. Assume thatA,B andA∪B are (i, j)−mX−β−closed. By Theorem 2 (1),mij
XExtB(A) =

X\A,mij
XExtB(B) = X\B and mij

XExtB(A ∪B) = X\(A ∪B). Furthermore,

mij
XExtB(A ∪B) = X\(A ∪B)

= (X\A) ∩ (X\B)
= mij

XExtB(A) ∩mij
XExtB(B).

Next, we give some notions for the product of two biminimal structure space as the
following:

Definition 6. Let (X,m1
X ,m

2
X) and (Y,m1

Y ,m
2
Y ) be biminimal structure spaces, A,B be

subsets of X and Y , respectively. Then,
mij

X×Y ClB(A×B) = [mij
XClB(A)× Y ] ∩ [X ×mij

Y ClB(B)]
where i, j = 1, 2 and i 6= j.

From Example 1, let Y = {a, b, c}, we have m1
Y = {∅, {a}, {b}, {a, c}, Y } and m2

Y =
{∅, {c}, {a, b}{b, c}, Y }. Therefore,
m12

X×Y ClB({2} × {c}) = [m12
XClB({2})× Y ] ∩ [X ×m12

Y ClB({c})] = {2, c}.

Lemma 8. Let (X,m1
X ,m

2
X) and (Y,m1

Y ,m
2
Y ) be biminimal structure spaces, A,B be

subsets of X and Y, respectively. Then,
mij

X×YExtB(A×B) = [mij
XExtB(A)× Y ] ∪ [X ×mij

YExtB(B)].

Proof. Let (Y,m1
Y ,m

2
Y ) be a biminimal structure subspace of (X,m1

X ,m
2
X) and A a

subset of Y . Let us consider:
mij

X×YExtB(A×B) = (X × Y )\mij
X×Y ClB(A×B)

= (X × Y )\((mij
XClB(A)× Y ) ∩ (X ×mij

Y ClB(B)))

= ((X × Y )\(mij
XClB(A)× Y )) ∪ ((X × Y )\(X ×mij

Y ClB(B)))

= ((X\mij
XClB(A))× Y ) ∪ (X × (Y \mij

Y ClB(B))))

= (mij
XExtB(A)× Y ) ∪ (X ×mij

YExtB(B)).

The next results study the biminimal structure subspace and obtain some properties
of them. Furthermore, the product of the (i, j)−mX − β−exterior sets is introduced.



REFERENCES 921

Lemma 9. Let (W,m1
W ,m

2
W ) be a biminimal structure subspace of (X,m1

X ,m
2
X), A and

B are subsets of X and W, respectively, and A = B ∩W are (i, j) − mX − β − closed.
Then, mij

WExtB(A) = mij
XExtB(B) ∩W.

Proof. Let (W,m1
W ,m

2
W ) be a biminimal structure subspace of (X,m1

X ,m
2
X) and A

be a subset of Y . Consider,
mij

YExtB(A) = mij
WExtB(B ∩W )

= mij
XExtB(B ∩W ) ∩ Y

= [mij
XExtB(B) ∪mij

XExtB(W )] ∩ Y
= [mij

XExtB(B) ∩W ] ∪ [mij
XExtB(W ) ∩W ]

= [mij
XExtB(B) ∩W ].

4. Conclusion

This study investigated (i, j) −mX − β−exterior sets in a biminimal structure space
(BSS) and a biminimal structure subspace (BSs). First, (i, j)−mX − β−exterior sets in
a biminimal structure space are defined in Definition 5. Second, Theorem 1 present the
notion for the subsets of (i, j)−mX −β−exterior sets. Third, the relation of (i, j)−mX −
β−exterior sets and (i, j)−mX − β−closed and open sets are covered in Theorem 2. The
union, intersection and product of (i, j) −mX − β−exterior sets are given in Theorem 3
and Lemma 8. The authors describe the (i, j) − mX − β−exterior sets of a biminimal
structure subspace (BSs), in the last Section 3.

Acknowledgements

The authors thank the referees for valuable comments and suggestions on this manuscript.
This research was supported by the Faculty of Sciences and Engineering, Kasetsart Uni-
versity, Chalermprakiat Sakon Nakhon Province Campus, Thailand.

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