Journal of Indian Acad. Math.  
Vol. 47, No. 1 (2025) pp. 31-40  
ISSN: 0970-5120  
Purva Rajwade1  
A NOTE ON NANO FUZZY CLOSURE  
AND BICLOSURE SPACES  
and  
Rachna Navalakhe2  
Abstract: The aim of this paper is to present, clarify and analyze Nano  
fuzzy closure spaces and Nano fuzzy bi-closure spaces in relation to Nano  
fuzzy topological spaces. We have tried to analyze the basic properties of  
these new types of spaces.  
Keywords: Nano Fuzzy Topological Spaces, Nano Fuzzy Closure Space,  
Nano Fuzzy Biclosure Space.  
Mathematics Subject Classification (2010) No.: 03E72, 54A05, 54A40.  
1. Introduction  
Thivagar L. at al. [9, 6] introduced the concept of Nano topological spaces  
which were defined in terms of lower approximation, upper approximation and  
boundary region of a subset of a universe ꢀ using an equivalence relation on it and  
also defined Nano closed sets, Nano interior and Nano closure. Further,  
Bhuvneshwari K. et al. [2] introduced Nano generalized closed set in Nano  
topological spaces in 2014. B. A. Deole [5] has introduced Nano closure and Nano  
biclosure spaces in Nano topological spaces.  
After the theory of fuzzy sets, given by L. Zadeh [11], fuzzfication of  
topological spaces was done. This work is done by C. L. Chang [4] and defined fuzzy  
topological spaces.  
32  
PURVA RAJWADE AND RACHNA NAVALAKHE  
R. Navalakhe et al. [7] defined Nano fuzzy topological spaces with respect to  
a fuzzy subset ꢁ of an universe which is defined in terms of lower and upper  
approximations of ꢁ and studied Nano fuzzy closure and Nano fuzzy interior of a  
fuzzy subset. In this article we have presented the idea of Nano fuzzy closure spaces  
and Nano fuzzy bi-closure spaces and examined their characteristics.  
2. Preliminaries  
In this section we have narrated some of the important definition and results  
which are helpful in defining Nano fuzzy closure and Nano fuzzy bi-closure spaces  
in Nano fuzzy topological spaces.  
Definition 2.1 [5]: Let U be a non-empty finite set of objects called the  
universe and R be an equivalence relation on U and ꢂ ⊆ ꢀ. Then Nano closure  
operator is a function: ꢃꢄꢅꢆ: ꢇ(ꢂ) → ꢇ(ꢂ) such that for all ꢈ ⊆ ꢂ  
ꢊꢋꢆ(ꢂ) ꢌꢍ ꢈ ⊆ ꢊꢋꢆ(ꢂ)  
ꢎꢋꢆ(ꢂ) ꢌꢍ ꢈ ⊆ ꢎꢋꢆ(ꢂ)  
ꢃꢄꢅꢆ = ꢉ  
ꢂ; ꢏꢐ ꢑꢒꢓꢌꢔꢑ ꢕꢖꢗ ꢘ ꢌꢍ ꢈ = ꢘ  
where ꢊꢋ’s are elements of ꢊꢆ(ꢂ) and ꢎꢋ’s are elements of ꢎꢆ(ꢂ). Which satisfies  
three conditions:  
1. ꢃꢄꢅꢆ(ꢘ)= ꢘ  
2. ꢈ ⊆ ꢃꢄꢅꢆ(ꢈ)  
(
)
3. ꢃꢄꢅꢆ ꢈ ꢎ = ꢃꢄꢅꢆ(ꢈ) ꢃꢄꢅꢆ(ꢎ)  
Hence, (ꢂ, ꢃꢄꢅꢆ) is called Nano closure space.  
Definition 2.2 [7, 8]: Let ꢀ be the universe, ꢚꢛ and ꢚꢜ be equivalence  
relations on ꢀ. P1 and P2 are subsets of ꢀ. Then ꢝꢆꢛ(ꢇ ) and ꢝꢆꢜ(ꢇꢜ) satisfies the  
ꢛ
following axioms:  
1. U and ∈ꢝꢆꢛ(ꢇ ) and ꢝꢆꢜ(ꢇꢜ).  
ꢛ
2. The union of the elements of any sub collection of ꢝꢆꢛ(ꢇ ) is in ꢝꢆꢛ(ꢇ )  
ꢛ
ꢛ
and ꢝꢆꢜ(ꢇꢜ) is in ꢝꢆꢜ(ꢇꢜ).  
3. The intersection of the elements of any finite sub collection of ꢝꢆꢛ(ꢇ ) is  
ꢛ
in ꢝꢆꢛ(ꢇ ) and ꢝꢆꢜ(ꢇꢜ) is in ꢝꢆꢜ(ꢇꢜ).  
ꢛ
A NOTE ON NANO FUZZY CLOSURE AND BICLOSURE SPACES  
33  
Hence, ꢝꢆꢛ(ꢇ ) and ꢝꢆꢜ(ꢇꢜ) are called the Nano (ꢝꢛ, ꢝꢜ) bitopology on ꢀ  
ꢛ
( )  
ꢂ ) is called Nano (ꢝꢛ, ꢝꢜ) bitopological space.  
with respect to ꢇ and ꢇꢜ, (ꢀ, ꢝꢆ  
ꢛ
ꢞ,ꢟ  
ꢂ are known as Nano (1,2) open sets in ꢀ and elements of  
( )  
Elements of the ꢝꢆ  
ꢞ,ꢟ  
ꢠ
( )  
[ꢝꢆ  
ꢞ,ꢟ  
ꢂ ] are called Nano (1,2) closed sets.  
( )  
ꢂ ) is a Nano bitopological space with respect  
Definition 2.3 [1]: If (ꢀ, ꢝꢆ  
ꢞ,ꢟ  
to ꢂ where ꢂ ⊆ ꢀ and if ꢈ ⊆ ꢀ, then  
1. The Nano (1,2) closure of ꢈ is defined as the intersection of all Nano  
( )  
(1,2) closed sets containing ꢈ and it is denoted by ꢃꢝꢆꢞ,ꢟꢄꢅ ꢈ . It is the smallest  
Nano (1,2) closed set containing ꢈ.  
(
)
2. The Nano (1,2) interior of ꢈ is defined as the union of all Nano 1,2  
( )  
open subsets of ꢈ contained in ꢈ and it is denoted by ꢃꢝꢆꢞ,ꢟꢡꢖꢐ ꢈ . It is the largest  
Nano (1,2) open subset of ꢈ.  
Definition 2.4 [5]: Let U be a non-empty finite set of objects called the  
universe and ꢚꢛ and ꢚꢜ be two equivalence relations on U and ꢂ ⊆ ꢀ. Then Nano  
{
}
closure operator is a function: ꢃꢄꢅꢆ : ꢇ(ꢂ) → ꢇ(ꢂ) where ꢌ = 1,2 such that for all  
ꢢ
ꢈ ⊆ ꢂ  
ꢊꢋꢆ(ꢂ) ꢌꢍ ꢈ ⊆ ꢊꢋꢆ(ꢂ)  
ꢎꢋꢆ(ꢂ) ꢌꢍ ꢈ ⊆ ꢎꢋꢆ(ꢂ)  
ꢃꢄꢅꢆ = ꢉ  
ꢢ
ꢂ; ꢏꢐ ꢑꢒꢓꢌꢔꢑ ꢕꢖꢗ ꢘ ꢌꢍ ꢈ = ꢘ  
where ꢊꢋ’s are elements of ꢊꢆ(ꢂ) and ꢎꢋ’s are elements of ꢎꢆ(ꢂ). Which satisfies  
three conditions:  
( )  
( )  
1. ꢃꢄꢅꢆ ꢘ = ꢘ and ꢃꢄꢅꢆ ꢘ = ꢘ  
ꢞ
ꢟ
2. ꢈ ⊆ ꢃꢄꢅꢆ (ꢈ) and ꢈ ⊆ ꢃꢄꢅꢆꢟ(ꢈ)  
ꢞ
(
)
3. ꢃꢄꢅꢆ ꢈ ꢎ = ꢃꢄꢅꢆ (ꢈ) ꢃꢄꢅꢆꢞ(ꢎ) and  
ꢞ
ꢞ
(
)
ꢃꢄꢅꢆ ꢈ ꢎ = ꢃꢄꢅꢆ (ꢈ) ꢃꢄꢅꢆꢟ(ꢎ).  
ꢟ
ꢟ
That is there are two closure spaces (ꢂ, ꢃꢄꢅꢆ ) and (ꢂ, ꢃꢄꢅꢆꢟ). Hence,  
ꢞ
(ꢂ, ꢃꢄꢅꢆ , ꢃꢄꢅꢆꢟ) is called Nano biclosure space.  
ꢞ
34  
PURVA RAJWADE AND RACHNA NAVALAKHE  
Definition 2.5 [5]: Let (ꢂ, ꢃꢄꢅꢆ , ꢃꢄꢅꢆꢟ) be a Nano biclosure space. A Nano  
ꢞ
biclosure space (ꢣ, ꢃꢄꢅꢆ , ꢃꢄꢅꢆꢥ) is called a Nano biclosure subspace of  
ꢤ
{
}
{
}
(ꢂ, ꢃꢄꢅꢆ , ꢃꢄꢅꢆ ) if ꢣ ⊆ ꢂ and ꢃꢄꢅꢆ = ꢃꢄꢅꢆ  
ꢢ
ꢣ for each ꢌ = 1,2 , ꢧ = 3,4 and  
ꢞ
ꢟ
ꢦ
each subset ꢈ ⊆ ꢣ.  
Definition 2.6 Properties of Fuzzy Approximation Space  
[3, 10]: Let ꢚ be an arbitrary relation from ꢂ to ꢣ. The lower and upper  
ꢨ
approximation operators of a fuzzy set R and R satisfies the following properties: for  
( )  
all ꢩ, ꢪ ∈ ꢫ ꢂ ,  
ꢠ
ꢠ
(FL1)  
(FU1)  
(FL2)  
(FU2)  
( )  
(
)
ꢚ ꢩ = (ꢚ ꢩ )  
ꢠ
ꢠ
( )  
(
)
ꢚ ꢩ = (ꢚ ꢩ )  
(
)
ꢚ ꢩ ꢪ = ꢚ(ꢩ) ꢚ(ꢪ)  
ꢚ(ꢩ ꢪ) = ꢚ(ꢩ) ꢚ(ꢪ)  
ꢩ ≤ ꢪ ⇒ ꢚ(ꢩ) ≤ ꢚ(ꢪ)  
ꢩ ≤ ꢪ ⇒ ꢚ(ꢩ) ≤ ꢚ(ꢪ)  
(FL3)  
(FU3)  
(FL4)  
(FU4)  
(
)
( )  
ꢚ ꢩ ꢪ = ꢚ(ꢩ) ꢚ ꢪ  
ꢚ(ꢩ ꢪ) = ꢚ(ꢩ) ꢚ(ꢪ)  
Definition 2.7 [7]: Let ꢂ be a non-empty finite set, ꢚ be an equivalence  
( ) ( ) ( ) ( )  
relation on X, ꢁ ≤ ꢂ be a fuzzy subset and ꢝꢆ ꢁ = ꢬ 1ꢭ, 0ꢭ, ꢚ ꢁ , ꢚ ꢁ , ꢎꢗ ꢁ ꢮ.  
( )  
Then by property (2.6), ꢝꢆ ꢁ atisfies the following axioms  
( )  
ꢁ where 0ꢭ: ꢁ → ꢡ denotes the null fuzzy sets and 1ꢭ: ꢁ → ꢡ  
i.  
0ꢭ, 1ꢭ ∈ ꢝ(  
ꢆ
)
denotes the whole fuzzy set.  
( )  
ꢁ is a member of ꢝ(  
( )  
ꢁ .  
)
ii.  
Arbitrary union of members of ꢝ(  
)
ꢆ
ꢆ
( )  
ꢁ is a member of ꢝ(  
( )  
ꢁ .  
)
iii.  
Finite intersection of members of ꢝ(  
ꢆ
)
ꢆ
( )  
ꢁ is a topology on ꢂ called the Nano fuzzy topology on ꢂ with  
That is, ꢝ(  
)
ꢆ
respect to ꢁ.  
A NOTE ON NANO FUZZY CLOSURE AND BICLOSURE SPACES  
35  
( )  
ꢁ ) as the Nano fuzzy topological space (NFTS). The  
We call (ꢂ, ꢝ(  
)
ꢆ
( )  
ꢁ , are called Nano fuzzy  
elements of the Nano fuzzy topological space that is ꢝ(  
)
ꢆ
ꢠ
( )  
open sets and elements of [ꢝ(  
ꢁ ] are called Nano fuzzy closed sets.  
)
ꢆ
( )  
ꢁ ) be a Nano fuzzy topological space with  
Definition 2.8 [7]: Let (ꢂ, ꢝ(  
)
ꢆ
respect to ꢁ where ꢁ ≤ ꢂ and if ꢯ ≤ ꢂ then the Nano fuzzy interior of ꢯ is defined  
as union of all Nano fuzzy open subsets of ꢯ and it is denoted by ꢃꢍꢡꢖꢐ(ꢯ). That is,  
it is the largest Nano fuzzy open subset contained in ꢯ.  
Similarly, the Nano fuzzy closure of ꢯ is defined as the intersection of all  
Nano fuzzy closed sets containing ꢯ. It is denoted by ꢃꢍꢰꢅ(ꢯ) and it is the smallest  
Nano fuzzy closed set containing ꢯ.  
Definition 2.9 [1]:  
equivalence relations on  
( )  
Let ꢂ be a non-empty finite set, ꢚꢛ and ꢚꢜ be  
X, ꢁꢛ, ꢁꢜ ≤ ꢂ be fuzzy subsets and  
( )  
(
)
ꢝꢛ,ꢜ ꢁ = {ꢝꢛ ꢁꢛ , ꢝꢜ (ꢁꢜ)}. Then ꢝꢛ,ꢜ ꢁ satisfies the following axioms:  
ꢱ
ꢱ
ꢱ
ꢱ
(
)
1. 0ꢭ , 1ꢭ ∈ ꢝꢛ ꢁꢛ where 0ꢭ : ꢁꢛ → ꢡ denotes the null fuzzy sets and  
ꢞ
ꢞ
ꢱ
ꢞ
1ꢭ : ꢁꢛ → ꢡ denotes the whole fuzzy set and 0ꢭ , 1ꢭ ∈ ꢝꢜ (ꢁꢜ) where  
ꢞ
ꢟ
ꢟ
ꢱ
0ꢭ : ꢁꢜ → ꢡ denotes the null fuzzy sets and 1ꢭ : ꢁꢜ → ꢡ denotes the whole  
ꢟ
ꢟ
fuzzy set.  
(
)
(
)
(
)
2. Arbitrary union of members of ꢝꢛꢆ ꢁꢛ and ꢝꢜꢆ ꢁꢜ are in ꢝꢛꢆ ꢁꢛ and  
(
)
ꢝꢜꢆ ꢁꢜ respectively.  
(
)
(
)
(
)
3. Finite intersection of members of ꢝꢛꢆ ꢁꢛ and ꢝꢜꢆ ꢁꢜ are in ꢝꢛꢆ ꢁꢛ and  
(
)
ꢝꢜꢆ ꢁꢜ respectively.  
(
)
(
)
( )  
ꢁ
That is, ꢝꢛꢆ ꢁꢛ and ꢝꢜꢆ ꢁꢜ are called the Nano fuzzy bitopology ꢝꢛ,ꢜ  
ꢱ
( )  
on ꢂ with respect to ꢁꢛ ꢕꢖꢗ ꢁꢜ. We call ꢲꢂ, ꢝꢛ,ꢜ ꢁ ꢳ as the Nano fuzzy  
ꢱ
bitopological space (NFBTS). The elements of the Nano fuzzy bitopological space  
ꢠ
( )  
are called Nano fuzzy (1,2) open sets and elements of [ꢝꢛ,ꢜ ꢁ ] are called Nano  
ꢱ
fuzzy (1,2) closed sets.  
3. Nano fuzzy Closure Spaces  
Definition 3.1: Let ꢂ be a non-empty finite set of objects which called the  
universe and ꢚ be an equivalence relation defined on ꢂ and ꢁ be an fuzzy subset of  
ꢂ. Then Nano fuzzy closure operator is a function ꢃꢍꢰꢅꢆ: ꢫ(ꢂ) → ꢫ(ꢂ) where ꢫ(ꢂ)  
is the set of all fuzzy subsets of ꢂ, such that for all ꢯ ≤ ꢁ  
36  
PURVA RAJWADE AND RACHNA NAVALAKHE  
( )  
( )  
ꢚꢋ ꢁ ꢌꢍ ꢯ ≤ ꢚꢋ ꢁ  
ꢃꢍꢰꢅꢆ = ꢉ  
( )  
( )  
ꢎꢗꢋ ꢁ ꢌꢍ ꢯ ≤ ꢎꢗꢋ ꢁ  
ꢁ; ꢏꢐ ꢑꢒꢓꢌꢔꢑ ꢕꢖꢗ 0ꢭ ꢌꢍ ꢯ = 0ꢭ  
( )  
Where ꢚꢋ’s are elements of ꢚ ꢁ and ꢎꢗꢋ’s are elements of ꢎꢗ(ꢁ). Which  
satisfies three conditions:  
(
)
1. ꢃꢍꢰꢅꢆ 0ꢭ = 0ꢭ  
2. ꢯ ≤ ꢃꢍꢰꢅꢆ(ꢯ)  
(
)
3. ꢃꢍꢰꢅꢆ ꢯ ꢪ = ꢃꢍꢰꢅꢆ(ꢯ) ꢃꢍꢰꢅꢆ(ꢪ)  
Hence, (ꢂ, ꢃꢍꢰꢅꢆ) is called Nano fuzzy closure space.  
Definition 3.2: The elements of Nano fuzzy closure space are called Nano  
fuzzy open sets in Nano fuzzy closure spaces. The complement of Nano fuzzy open  
sets is called Nano fuzzy closed sets with respect to the Nano fuzzy closure space.  
Definition 3.3: A fuzzy subset ꢩ of a Nano fuzzy closure space (ꢂ, ꢃꢍꢰꢅꢆ)  
( )  
is called Nano fuzzy closed if ꢴꢃꢍꢰꢅꢆ ꢩ ꢵ = ꢩ.  
The complement of Nano fuzzy closed set is called Nano fuzzy open.  
4. Nano fuzzy Bi-closure Spaces  
Definition 4.1: Let ꢂ be a non-empty finite set of objects which is called the  
universe and ꢚꢛ and ꢚꢜ be two equivalence relations on ꢂ and ꢁ be any fuzzy subset  
of ꢂ. Then Nano fuzzy closure operator is a function: ꢃꢍꢰꢅꢆ : ꢫ(ꢂ) → ꢫ(ꢂ), where  
ꢢ
{
}
ꢌ = 1,2 , and ꢫ(ꢂ) is the set of all fuzzy subsets of ꢂ, such that for all ꢯ ≤ ꢁ  
( )  
( )  
ꢚꢋ ꢁ ꢌꢍ ꢯ ≤ ꢚꢋ ꢁ  
ꢃꢍꢰꢅꢆ = ꢉ  
( )  
( )  
ꢎꢗꢋ ꢁ ꢌꢍ ꢯ ≤ ꢎꢗꢋ ꢁ  
ꢁ; ꢏꢐ ꢑꢒꢓꢌꢔꢑ ꢕꢖꢗ 0ꢭ ꢌꢍ ꢯ = 0ꢭ  
( )  
Where ꢚꢋ’s are elements of ꢚ ꢁ and ꢎꢗꢋ’s are elements of ꢎꢗ(ꢁ). Which  
satisfies three conditions:  
(
)
(
)
1. ꢃꢍꢰꢅꢆ 0ꢭ = 0ꢭ and ꢃꢍꢰꢅꢆ 0ꢭ = 0ꢭ  
ꢞ
ꢟ
2. ꢯ ≤ ꢃꢍꢰꢅꢆ (ꢯ) and ꢯ ≤ ꢃꢍꢰꢅꢆꢟ(ꢯ)  
ꢞ
A NOTE ON NANO FUZZY CLOSURE AND BICLOSURE SPACES  
37  
(
)
3. ꢃꢍꢰꢅꢆ ꢯ ꢪ = ꢃꢍꢰꢅꢆ (ꢯ) ꢃꢍꢰꢅꢆꢞ(ꢪ)  
ꢞ
ꢞ
and  
(
)
ꢃꢍꢰꢅꢆ ꢯ ꢪ = ꢃꢍꢰꢅꢆ (ꢯ) ꢃꢍꢰꢅꢆꢟ(ꢪ)  
ꢟ
ꢟ
That is there are two fuzzy closure spaces (ꢂ, ꢃꢍꢰꢅꢆ ) and (ꢂ, ꢃꢍꢰꢅꢆꢟ).  
ꢞ
Hence, (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) is called Nano fuzzy biclosure space.  
ꢞ
Definition 4.2: The elements of Nano fuzzy biclosure space are called Nano  
fuzzy open sets in Nano fuzzy bi-closure spaces. The complement of Nano fuzzy  
open sets is called Nano fuzzy closed sets with respect to the Nano fuzzy biclosure  
space.  
Definition 4.3: Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) be a Nano fuzzy biclosure space. A  
ꢞ
Nano fuzzy biclosure space (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ) is called a Nano fuzzy biclosure  
ꢤ
subspace of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ  
)
if ꢣ ⊆ ꢂ and ꢃꢍꢰꢅꢆ = ꢃꢍꢰꢅꢆ ꢣ for each  
ꢞ
ꢟ
ꢦ
ꢢ
{
}
{
}
ꢌ = 1,2 , ꢧ = 3,4 and each fuzzy subset ꢁ ≤ ꢣ.  
Remark 4.4: 1. Nano fuzzy open sets of Nano fuzzy bi-closure space are  
open in both Nano fuzzy closure spaces.  
2. A fuzzy subset ꢩ of a Nano fuzzy bi-closure space (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) is  
ꢞ
( )  
called Nano fuzzy closed if ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢩ ꢳ = ꢩ.  
ꢞ
ꢟ
The complement of Nano fuzzy closed set is called Nano fuzzy open.  
3. ꢩ is a Nano fuzzy closed subset of Nano fuzzy biclosure space  
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) if and only if ꢩ is Nano fuzzy closed subset of both  
ꢞ
(ꢂ, ꢃꢍꢰꢅꢆ ) and (ꢂ, ꢃꢍꢰꢅꢆꢟ).  
ꢞ
4. Let ꢩ be a Nano fuzzy closed subset of a Nano fuzzy biclosure space  
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ  
)
ꢞ
ꢟ
The following conditions are equivalent.  
( )  
1. ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢩ ꢳ = ꢩ  
ꢟ
ꢞ
( )  
( )  
2. ꢃꢍꢰꢅꢆ ꢩ = ꢩ, ꢃꢍꢰꢅꢆ ꢩ = ꢩ  
ꢞ
ꢟ
38  
PURVA RAJWADE AND RACHNA NAVALAKHE  
Remark 4.5: Let ꢩ be a fuzzy subset of a Nano fuzzy biclosure space  
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ ). If ꢩ is a Nano fuzzy open set in (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ), then  
ꢞ
ꢟ
ꢞ
(
)
(
)
ꢩ ꢳ  
ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢩ ꢳ = ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢞ
ꢟ
ꢟ
ꢞ
Proposition 4.6: Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) be a Nano fuzzy biclosure space  
ꢞ
and let ꢩ ≤ ꢂ. Then  
(
)
1. ꢩ is Nano fuzzy open if and only if ꢩ = 1ꢭ  
ꢶꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢩ ꢳꢷ  
ꢞ
ꢟ
(
)
2. If ꢩ is Nano fuzzy open and ꢩ ≤ ꢯ, then ꢩ ≤ 1ꢭ  
ꢶꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢩ ꢳꢷ.  
ꢞ
ꢟ
Proof: 1. Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) be a Nano fuzzy biclosure space and let  
ꢞ
ꢯ ≤ ꢂ and ꢯ is Nano fuzzy open then 1ꢭ  
ꢩ is Nano fuzzy closed in Nano fuzzy  
(
)
biclosure space. So, by definition, ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢩ ꢳ = 1ꢭ  
ꢩ. This  
ꢞ
ꢟ
(
)
implies that ꢩ = 1ꢭ  
ꢶꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ 1ꢭ  
ꢩ ꢳꢷ.  
ꢞ
ꢟ
2. By part (1) obvious.  
Proposition 4.7: Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) is a Nano fuzzy biclosure space.  
ꢞ
If ꢩ and ꢪ are two Nano fuzzy closed subsets of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ). Then ꢩ ꢪ is  
ꢞ
also Nano fuzzy closed in (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ).  
ꢞ
Proposition 4.8: Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) is a Nano fuzzy biclosure space.  
ꢞ
If ꢩ and ꢪ are two Nano fuzzy closed subsets of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) then ꢩ ꢪ is  
ꢞ
Nano fuzzy closed if ꢃꢍꢰꢅꢆ and ꢃꢍꢰꢅꢆ are disjoint.  
ꢞ
ꢟ
Proof: Let  
ꢩ
and  
ꢪ
are two Nano fuzzy closed subsets of  
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ).  
ꢞ
( )  
( )  
Then ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢩ ꢳ = ꢩ and ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢪ ꢳ = ꢪ  
ꢞ
ꢟ
ꢞ
ꢟ
Now,  
(
)
ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢩ ꢪ ꢳ = ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢴꢩ) ꢃꢍꢰꢅꢆꢟ(ꢪ)ꢵꢳ  
ꢞ
ꢟ
ꢞ
ꢟ
( )  
( )  
= ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢩ ꢳ ꢃꢍꢰꢅꢆ ꢲꢃꢍꢰꢅꢆ ꢪ ꢳ = ꢩ ꢪ  
ꢞ
ꢟ
ꢞ
ꢟ
A NOTE ON NANO FUZZY CLOSURE AND BICLOSURE SPACES  
39  
Therefore ꢩ ꢪ is Nano closed if ꢃꢍꢰꢅꢆ and ꢃꢍꢰꢅꢆ are disjoint.  
ꢞ
ꢟ
Proposition 4.9: If (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ) is a Nano fuzzy biclosure subspace  
ꢤ
of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ), then for every Nano fuzzy open subset ꢸ of  
ꢞ
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ ), ꢸ ꢣ is an Nano fuzzy open set in (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ).  
ꢞ
ꢟ
ꢤ
Proof: Let ꢸ be a Nano fuzzy open set in (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ), then by  
ꢞ
property we can say that ꢸ is Nano fuzzy open in both ꢃꢍꢰꢅꢆ and ꢃꢍꢰꢅꢆ .  
ꢞ
ꢟ
Thus,  
(
ꢣ
)
(
ꢣ
)
(
ꢂ
)
(
)
ꢃꢍꢰꢅꢆ  
ꢸ ꢣ = ꢃꢍꢰꢅꢆ  
ꢸ ꢣ  
ꢣ ≤ ꢃꢍꢰꢅꢆ  
ꢢ
ꢸ
ꢣ = ꢂ  
ꢸ
ꢣ =  
ꢦ
ꢢ
ꢣ
(ꢸ ꢣ) for each ꢌ = {1,2}, ꢧ = {3,4}. Consequently, ꢸ ꢣ is Nano fuzzy open in  
both (ꢣ, ꢃꢍꢰꢅꢆ and (ꢣ, ꢃꢍꢰꢅꢆꢥ). Therefore, ꢸ ꢣ is Nano fuzzy open in  
)
ꢤ
(ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ).  
ꢤ
Proposition 4.10: Let (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ) be a Nano fuzzy biclosure space  
ꢞ
and let (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ  
)
be a Nano fuzzy biclosure subspace of  
ꢤ
ꢥ
(ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆ ). If ꢩ is a Nano fuzzy closed subset of (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ),  
ꢞ
ꢟ
ꢤ
then ꢩ is also a Nano fuzzy closed subset of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ).  
ꢞ
Proof: Let ꢩ be a Nano fuzzy closed subset of (ꢣ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢥ). Then  
ꢤ
ꢃꢍꢰꢅꢆ (ꢩ) = ꢩ and ꢃꢍꢰꢅꢆꢥ(ꢩ) = ꢩ. Since ꢩ is Nano fuzzy closed subset of both  
ꢤ
(ꢂ, ꢃꢍꢰꢅꢆ ) and (ꢂ, ꢃꢍꢰꢅꢆꢟ).  
ꢞ
Consequently, ꢩ is a Nano fuzzy closed subset of both (ꢂ, ꢃꢍꢰꢅꢆꢞ) and  
(ꢂ, ꢃꢍꢰꢅꢆ ). Therefore, ꢩ is a Nano fuzzy closed subset of (ꢂ, ꢃꢍꢰꢅꢆ , ꢃꢍꢰꢅꢆꢟ).  
ꢟ
ꢞ
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1, 2. Department of Applied Mathematics  
& Computational Science,  
(Received, October 7, 2024)  
Shri G. S. Institute of Technology & Science, Indore (M.P.)  
1. E-mail: rajwadepurva@gmail.com  
2. E-mail: sgsits.rachna@gmail.com