Ratio Mathematica Volume 44, 2022

Remarks on Interiors and Closures of Weak
Open Sets in Bigeneralized Topological

Spaces

M. Anees Fathima*
R. Jamuna Rani†

Abstract

We establish the relationships between the interior and closure opera-
tors among the µij-semiopen, µij-preopen, αµij-open, βµij-open sets
in bigeneralized topological spaces.
Keywords: Generalized topology, bigeneralized topology, µi-open
set, µi-closed set.
12020 AMS subject classifications: 54A05, 54A10.

*Research Scholar(Reg.No: 19121172092012), PG and Research Department of Mathematics,
Rani Anna Government College for Women, Affiliated to Manonmaniam Sundaranar University,
Abishekapatti, Tirunelveli-627012, Tamilnadu, India. Email: afiseyan09@gmail.com.

†Assistant Professor, PG and Research Department of Mathematics, Rani Anna Govern-
ment College for Women, Affiliated to Manonmaniam Sundaranar University, Abishekapatti,
Tirunelveli-627012, Tamilnadu, India. Email:jamunarani1977@gmail.com.
1

1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence
agreement.

75

Received on June, 2022. Accepted on September, 2022. doi: 10.23755/rm.v39i0.892. ISSN:



M. Anees Fathima, R. Jamuna Rani

1 Introduction

The study of generalized topological spaces (briefly GTS) was first initiated
by A.Csaszar on 2002 (Császár, 2002). In that, he established the interior and
closure operators in GTS. Later in 2010 (Boonpok, 2010)this study is extended
to bigeneralized topological spaces (briefly BGTS) by C. Boonpak. In his pa-
per, he found the conception of (m, n)-closed and (m, n)-open sets in BGTS.
Also, he introduced (m, n)g-regular open, (m, n)g-semi open, (m, n)g-pre open,
(m, n)g − α-open . With these definitions, the properties of the weak open sets
were studied by A. Jamuna Rani and M. Anees Fathima in (Fathima and Rani,
2019; Rani and Fathima, 2020) and some of its characterizations are also anal-
ysed.

The purpose of this paper is to prove the relationships between the interior and
closure operators among µij-semi open, µij-preopen, αµij-open and βµij- open
sets in BGTS. Also we establish some of its Characterizations.

2 Preliminares

In this section, We provide some basic definitions and notations which are
most essential to understand the subsequent section.

For any non empty set X and ξ ∈ p(X), ξ is said to be GT (Császár, 2002) if
ϕ ∈ ξ and ξ is closed under arbitrary union. Also, the function γ ∈ Γ (where Γ
denotes the collection of all mappings γ : p(X) → p(X) possessing the property
of monotony. ie.,if A ⊂ B ⇒ γ(A) ⊂ γ(B)) is said to be µ-friendly (Császár,
2006)if γ(A) ∩ L ⊂ γ(A ∩ L) for every A ⊂ X and L ∈ µ. If γ ∈ Γ and
µ = {A ⊂ X/A ⊂ γ(A)}is the family of all γ- open sets, then µ is a GT (Császár,
2002) The pair (X, µ) is a called GTS. In (Sivagami, 2008) the family of all µ-
friendly functions is denoted by Γ4 and (X, γ) is called the γ-space. It is further
proved that every γ-space is a quasi-topological space in (Császár, 2008) and all
the results established in for γ-spaces are valid for quasi-topological spaces. Also
for γ ∈ Γ, define γ∗ : p(X) → p(X) by γ∗(A) = X − γ(X − A) (Császár, 1997)
for every subset A of X.

Let X be a non empty set and µ1, µ2 be generalized topologies on X. A triple
(X, µ1, µ2) is said to be a bigeneralized topological space. Let A be subset of a
bigeneralized topological space X. Then the closure of A and the interior of A
with respect to µm are denoted by cµm(A) and iµm(A) respectively, for m = 1, 2.
(Boonpok, 2010) A subset A of a bigeneralized topological space(X, µ1, µ2)
is said to be µij-semiopen (Fathima and Rani, 2019)(resp.µij-preopen (Rani and
Fathima, 2020) αµij-openβµij-open (Jamuna Rani and Anees Fathima, 2021))
if A ⊂ cµiiµj(A) where i, j = 1, 2 and i ̸= j (resp. A ⊂ iµicµj(A), A ⊂

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Remarks on Interiors and Closures of Weak Open Sets in Bigeneralized
Topological Spaces

iµicµjiµi(A),A ⊂ cµiiµjcµi(A)).

Proposition 1.1. (Min, 2009)Let (X, µ) be a generalized topological space. For
subsets A and B of X, the following properties holds.

(a) cµ(X − A) = X − iµ(A) and iµ(X − A) = X − cµ(A).

(b) If (X − A) ∈ µ, then cµ(A) = A and if A ∈ µ, then iµ(A) = A.

(c) If A ⊆ B, then cµ(A) ⊆ cµ(B) and iµ(A) ⊆ iµ(B).

(d) If A ⊆ cµ(A) and iµ(A) ⊆ A.

(e) cµ (cµ (A)) = cµ (A) and iµ (iµ (A)) = iµ (A).

Proposition 1.2. (Jamunarani et al., 2010) Let (X, γ) be a γ- space . Then G ∩
cγ(A) ⊂ cγ(G ∩ A), for every A ⊂ X and γ-open set G of X.

Proposition 1.3. Let (X, µ) be a quasi topological space and A, B ⊆ X, the
following holds.

(a) If A and B are µ-open sets, then A ∩ B is µ-open (Sivagami, 2008)

(b) iµ(A∩B) = iµ(A)∩iµ(B), for every subsets A and B of X (Császár, 2008)

(c) cµ(A ∪ B) = cµ(A) ∪ cµ(B), for every subsets A and B of X(Sivagami,
2008)

Proposition 1.4. (Fathima and Rani, 2019) Let (X, µ) be a generalized topologi-
cal space. Let A be a subset of X. Then the following hold.

(a) cσij(A) is the smallest µij-semi closed set containing A.

(b) A is µij-semi closed if and only if A = cσij(A).

(c) x ∈ cσij(A) if and only if for every µij-semi open G containing x, G ∩ A ̸=
ϕ.

(d) cσij ∈ Γ012+.

Proposition 1.5. (Fathima and Rani, 2019)Let (X, µ1, µ2) be a bigeneralized
topological space. Let A be a subset of X. Then the following hold.

(a)
(
iσij

)∗
= cσij .

(b)
(
cσij

)∗
= iσij .

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M. Anees Fathima, R. Jamuna Rani

(c) iσij(X − A) = X − cσij(A) for every subset A of X.

(d) cσij(X − A) = X − iσij(A) for every subset A of X.

Proposition 1.6. (Fathima and Rani, 2019)Let (X, µ1, µ2) be a bigeneralized
topological space. Let A be a subset of X. Then the following hold.

(a) A is µij-semi open if and only if A is cµiiµj - open if and only if A =
icµiiµj (A).

(b) iσij(A) = icµiiµj and cσij = ccµiiµj .

(c) iσij(A) = A ∩ cµiiµj(A).

(d) cσij(A) = A ∩ iµicµj(A).

Similar results from (Jamuna Rani and Anees Fathima, 2021; Rani and Fathima,
2020; Jamuna Rani and Anees Fathima, 2020) are also used in the next section.

3 Relationship between the operators:
The following theorem gives some of the relationships between iσij , cσij , iαij ,

cαij , iµi and cµi .

Theorem 1.1.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iσij(A) = A ∩ cµiiµj(A)

(b) cσij(A) = A ∪ iµicµj(A)

(c) iαij(A) = A ∩ iµicµjiµi(A)

(d) cαij(A) = A ∪ cµiiµjcµi(A)

(e) iµi(cσij(A)) = iµicµj(A)

(f) cµi(iσij(A)) = cµiiµj(A)

(g) cσij(iµi)(A) = iµicµjiµi(A)

(h) iσij(cµi)(A) = cµiiµjcµi(A)

(i) cσjiiσij(A) = (A ∩ cµiiµj(A)) ∪ iµjcµiiµj(A) for every subset A of X.

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Remarks on Interiors and Closures of Weak Open Sets in Bigeneralized
Topological Spaces

(j) iσijcσji(A) = (A ∩ cµiiµjcµi(A)) ∪ iµjcµi(A) for every subset A of X.

Proof. (a) By theorem 1.6(b), iσij = icµiiµj implies iσij(A) ⊂ A ∩ cµiiµj(A).
For the reverse part, A ∩ cµiiµj ⊂ cµiiµj(A) ⊂ cµiiµj

(
A ∩ cµiiµj(A)

)
. Thus

A ∩ cµiiµj = icµiiµj (A ∩ cµiiµj) ⊂ iσij(A).
(b) The result follows from (a).
(c) The proof is similar to the proof of (a).
(d) The result follows from (c).
(e) iµi

(
cσij(A)

)
= iµi

(
A ∪ iµicµj(A)

)
⊂ iµi

(
A ∪ cµj(A)

)
= iµicµj(A).

Also, by (b), iµicσij(A) ⊃ iµiiµicµj(A) = iµicµj(A).
(f) The result follows from (e).
(g) By (b), cσij(iµi(A)) = iµi(A) ∪ iµicµjiµi(A) = iµicµjiµi(A).
(h) By(a), iσij(cµi(A)) = cµi(A) ∩ cµiiµjcµi(A) = cµiiµjcµi(A).
(i) By(a),cσjiiσij(A) = (A ∩ cµiiµj(A)) ∪ iµjcµi(A ∩ cµiiµj(A)).
Here, cµi(A∩cµiiµj(A)) ⊂ cµiiµj(A) and cµi(A∩cµiiµj(A)) ⊃ cµi

(
iµj(A) ∩ cµiiµj(A)

)
= cµiiµj(A).
(j) Now, iσijcσji(A) = (A ∪ iµjcµi(A)) ∩ cµiiµj(A ∪ iµjcµi(A))
= (A ∪ iµjcµi(A)) ∩ cµjiµjcσji(A) =

(
A ∪ iµjcµi(A)

)
∩ cµiiµjcµi(A) by (e).

Hence iσijcσji(A) =
(
A ∩ cµiiµjcµi(A)

)
∪ iµjcµi(A). 2

The following theorem shows some results for the operators iπij , cπij , iβij , cβij .

Theorem 1.2.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iπij(A) = A ∩ iµicµj(A)

(b) cπij(A) = A ∪ cµiiµj(A)

(c) iβij(A) = A ∩ cµiiµjcµi(A)

(d) cβij(A) = A ∪ iµicµjiµi(A)

(e) iµi(iσji(A)) = iµi(A)

(f) cµi(cσji(A)) = cµi(A).

Proof. (a) In (Rani and Fathima, 2020), by theorem 3.2(e), we have iπij = iiµicµj
and so iπij(A) ⊂ A ∩ iµicµj(A). Let x ∈ iµicµj(A). Let G be any µi open set
containing x such that G ∩ iµicµj(A) is a µi -open set containing x. since x ∈
cµj(A) and G ∩ iµicµj(A) ∩ A ̸= ϕ and so x ∈ cµj

(
iµicµj(A) ∩ A

)
. Therefore,

iµicµj(A) ⊂ cµjiµicµj(A) ∩ A and so iµicµj(A) ⊂ iµicµj(A ∩ iµicµj(A)),
A ∩ iµicµj(A) ⊂ iµicµj(A) ⊂ iµicµj

(
A ∩ iµicµj(A)

)
, A ∩ iµicµj(A)

79



M. Anees Fathima, R. Jamuna Rani

= iiµicµj
(
A ∩ iµicµj(A)

)
⊂ iπij(A) ∩ iµicµj(A) ⊂ iπij(A). Hence iπij(A) =

A ∩ iµicµj(A).
(b) The result follows from (a).
(c) The proof is similar to the proof of (a).
(d) By theorem 3.10(i) in (Jamuna Rani and Anees Fathima, 2021), the proof is
similar to the proof of (b).
(e) iµi(iσji(A)) = iµi(A ∩ cµjiµi(A)) by theorem 1.1(a) and so iµi(iσji(A))
= iµi(A) ∩ iµicµjiµi(A) = iµi(A) by proposition 1.3(b).
(f) The result follows from (e). 2

The following theorem gives characterizations of µij-semi open, µij-preopen,
αµij-open, βµij-open sets using the interior and closure operators.

Theorem 1.3.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) A ∈ πij(µ) if and only if cσij(µ) = iµicµj(A)

(b) A is µij-preclosed if and only if iσij(A) = cµiiµj(A).

(c) A ∈ σij(µ) if and only if cπij(A) = cµiiµj(A).

(d) A is µij-semiclosed if and only if iπij(A) = iµicµj(A).

(e) A ∈ αij(µ)if and only if cβij(A) = iµicµjiµi(A).

(f) A is αµij -closed if and only if iβij(A) = cµiiµjcµi(A).

(g) A ∈ βij(µ) if and only if cαij(A) = cµiiµjcµi(A).

(h) A is βµij -closed if and only if iαij(A) = iµicµjiµi(A).

Proof. The proof follows from theorem 1.1(b),1.2(b),1.2(d), 1.1(d).2

Theorem 1.4.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) cµi(cπij(A)) = cµi(A).

(b) iµi(iπij(A)) = iµi(A).

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Remarks on Interiors and Closures of Weak Open Sets in Bigeneralized
Topological Spaces

(c) cµi(cβji(A)) = cµi(A).

(d) iµi(iβji(A)) = iµi(A).

(e) iµi(cπji(A)) ⊂ cµjiµi(A) and cµjiµi(cπji(A)) = cµjiµi(A).

(f) iµi(cπji(A)) = iµicµjiµi(A).

(g) cβij(iµi(A)) = iµicµjiµi(A) = iµi(cβij(A)).

(h) cβij(iβji(A)) = iβji(cβij(A)) =
(
A ∪ iµicµjiµi(A)

)
∩ cµjiµicµj(A).

(i) iβij (cµi(A)) = cµiiµjcµi(A) = cµiiβij(A).

(j) cσij
(
iσji(A)

)
⊂ iβji

(
cβij(A)

)
⊂ iσji

(
cσij(A)

)
.

Proof. (a) Clearly, cµi(A) ⊂ cµi
(
cπij(A)

)
.

Again, cµi
(
cπij(A)

)
⊂ cµi (cµi(A)) = cµi(A).

(b) The proof follows from (a).
The proof of (c) and (d) are similar to (a) and (b).
(e) Similar proof, so omitted.
(f) By (e), iµi

(
cπji(A)

)
⊂ iµicµjiµi(A). Again, iµi

(
cπji(A)

)
= iµi

(
A ∪ cµjiµi(A)

)
⊃

iµicµjiµi(A). Therefore, iµi
(
cπji(A)

)
= iµicµjiµi(A).

(g) By theorem 1.2(d), cβij (iµi(A)) = iµi(A) ∪ iµicµjiµi (iµi(A)) = iµicµjiµi(A).
Again, iµi

(
cβij(A)

)
= iµi

(
A ∪ iµicµjiµi(A)

)
⊃ iµicµjiµi(A).

For Converse part, iµi
(
cβij(A)

)
⊂ iµi

(
cπij(A)

)
⊂ cµjiµi(A) ⊂ iµicµjiµi(A).

(h) Similar proof, so omitted.
(i) The proof follows from (g).
(j) The proof follows from 1.1(i) and 1.1(j).2

The following theorem gives a characterization of βij(µ)-open sets.

Theorem 1.5.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following are equivalent.

(a) A ∈ βji(µ).

(b) A ⊂ iβji
(
cβij(A)

)
.

(c) A ⊂ iσji
(
cσij(A)

)
.

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M. Anees Fathima, R. Jamuna Rani

Proof. (a) ⇒ (b) If A ∈ βji(µ), then A =iβji(A) ⊂ iβji
(
cβij(A)

)
.

(b) ⇒ (c) By theorem 1.4(j), the proof follows.
(c) ⇒ (a) A ⊂ iσji

(
cσij(A)

)
⇒ A ⊂

(
cσij(A) ∩ cµjiµi

(
cσij(A)

))
=
(
cσij(A) ∩ cµjiµicµj(A)

)
by theorem 1.1(e) and so A ⊂ cµjiµicµj(A). 2

In the following theorem, we prove some relationships between iαij ,cαij with
iµi , cµi .

Theorem 1.6.
Let (X, µ1, µ2) be a bigenralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) cµicαij(A) = cαijcµi(A)= cµi(A).

(b) iµiiαij(A) = iαijiµi(A)= iµi(A).

(c) cαij
(
iµj(A)

)
= cµiiµj(A).

d) cµj
(
iαij(A)

)
= cµjiµi(A).

(e) iαij
(
cµj(A)

)
= iµicµj(A).

(f) iµj
(
cαij(A)

)
= iµjcµi(A).

Proof. (a) The proof follows from the theorem 1.1(d).
(b) The proof of (b) follows from (a).
(c) The proof follows from the theorem 1.1(d).
(d) cµj

(
iαij(A)

)
⊂ cµj

(
cµj(A) ∩ iµicµjiµi(A)

)
= cµjiµi(A).

Again, cµj
(
iαij(A)

)
⊃ cµj (A ∩ iµi(A)) = cµjiµi(A).

(e) The proof of (e) follows from (c).
(f) The proof of (f) follows from (d). 2

The following theorem shows some relationships between iαij , cαij with iσij ,
cσij , iπij ,cπij , iβij and cβij .

Theorem 1.7.
Let (X, µ1, µ2) be a bigenralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iαij
(
cσij(A)

)
= iµicµj(A).

(b) iαij
(
cπji(A)

)
= iµicµjiµi(A).

(c) iαij
(
cβji(A)

)
= iµicµjiµi(A).

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Remarks on Interiors and Closures of Weak Open Sets in Bigeneralized
Topological Spaces

(d) cαij
(
iσij(A)

)
=cµiiµj(A).

(e) cαij
(
iπji(A)

)
= cµiiµjcµi(A).

(f) cαij
(
iβji(A)

)
= cµiiµjcµi(A).

Proof. (a) The proof follows from the theorem 1.1(c), 1.1(f).
(b) The proof follows from the theorem 1.1(c), 1.4(g) and 1.1(a).
(c) The proof follows from the theorem 1.1(c), 1.4(h).
(d) The proof of (d) follows from (a).
(e) The proof of (e) follows from (b).
(f) The proof of (f) follows from (c). 2

Theorem 1.8.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iσijcαij(A) = cαij(A) ∩ cµiiµjcµi(A)

(b) cσijiαij(A) = iαij(A) ∩ iµicµjiµi(A)

(c) cσijcαji(A) =cαji(A) ∩ cσij(A).

(d) iσijiαji(A) =iαji(A) ∩ iσij(A).

(e) iσijiβij(A) = iσij(A).

(f) cσijcβij(A) = cσij(A).

Proof. (a) The result follows from the theorem 1.1(a) and 1.6(f).
(b) The proof of (b) follows from (a).
(c) The result follows from the theorem 1.1(b) and 1.6(a).
(d) The proof of (d) follows from (c).
(e) The result follows from the theorem 1.1(a), 1.4(d) and 1.2(c).
(f) The proof of (f) follows from (e).

Theorem 1.9.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iπijcαji(A) = cαji(A) ∩ iµicµj(A).

(b) iπijiβji(A) = iπij(A).

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M. Anees Fathima, R. Jamuna Rani

(c) cπijiαji(A) = iαji ∩ cµiiµj(A).

(d) cπijcβji(A) = cπij(A).

Proof. (a) The result follows from the theorem 1.2(a) and 1.6(a).
(b) The result follows from the theorem 1.2(a) and 1.4(i).
(c) The proof of (c) follows from (a).
(d) The proof of (d) follows from (b).2

Theorem 1.10.
Let (X, µ1, µ2) be a bigeneralized topological space. Let A be a subset of X and
µi ∈ Γ4. Then the following hold.

(a) iβijcαij(A) = cαij(A) ∩ cµiiµjcµi(A)= cµiiµjcµi(A).

(b) cβijiαij(A) = iµicµjiµi(A) = iαij(A) ∩ iµicµjiµi(A).

Proof. (a) The result follows from the theorem 1.2(c) and 1.6(a).
(b) The proof of (b) follows from (a). 2

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