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Fuzzy Double S – Hausdorff Spaces 
 

Sowmya.R#1, Dr.J.Srikiruthika#2 
#1 Research Scholar, Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for 

Women, Coimbatore, India. Email ID: 18phmap001@avinuty.ac.in 
#2 Assistant Professor of Mathematics, Department of Science and Humanities, School of Engineering, Avinashilingam 

Institute for Home Science and Higher Education for Women, Coimbatore, India. Email ID: kiruthika_sh@avinuty.ac.in 

 

Article History: 

Received: 25-07-2024 

Revised: 12-09-2024 

Accepted: 29-09-2024 

Abstract:  

This study introduces and explores the concept of fuzzy double s-Hausdorff spaces within 

the field of fuzzy topology. Fuzzy double s-Hausdorff spaces extend the classical Hausdorff 

condition by incorporating two distinct fuzzy topologies on a single set, allowing for a more 

nuanced approach to separation where uncertainty and gradation are involved. The research 

develops formal definitions and characterizations of these spaces, establishing key theorems 

that outline their properties and behavior. 

In this article, a meaning of FDS -T2 remains presented through covering the description of 

S- T2   introduce by Srivastava [3] and obtain result analogues towards the result of S- T2. 

Keywords: FDS, FDT, FDTS, FDS-T2. 

 

1. Introduction 

The concept of fuzzy set theory, introduced by Zadeh in 1965, has revolutionized various fields by 

allowing the representation of uncertainty and imprecision. In topology, this has led to the development 

of fuzzy topological spaces, which extend classical topological concepts to accommodate the 

vagueness inherent in many real-world scenarios. Among these, the notion of fuzzy s-Hausdorff 

spaces, which generalizes the classical Hausdorff condition, plays a crucial role in understanding 

separation axioms within a fuzzy context. 

The study of fuzzy topological spaces has been further enriched by the introduction of the double s-

Hausdorff space, a variant that incorporates two distinct Hausdorff-type conditions into the fuzzy 

framework. This development is essential for analyzing complex topological structures where 

traditional binary logic falls short. 

This article explores the fuzzy double s-Hausdorff space, focusing on its fundamental properties and 

implications in broader topological theory. We begin by revisiting the basic definitions and concepts 

related to fuzzy topologies, followed by an in-depth examination of double s-Hausdorff spaces. By 

establishing key theorems and proving their significance, we aim to provide a comprehensive 

understanding of this advanced topic in fuzzy topology. 

Objectives:  

• To formally define the concept of fuzzy double s-Hausdorff spaces and provide clear 

characterizations that distinguish them from other fuzzy topological spaces. 

• To investigate the fundamental properties of fuzzy double s-Hausdorff spaces, including their 

relationships with classical Hausdorff spaces and other fuzzy separation axioms.  

• To develop and prove key theorems related to fuzzy double s-Hausdorff spaces, contributing 

to the theoretical foundation of fuzzy topology. 



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Methods:  

 This study on fuzzy double s-Hausdorff spaces employs a combination of theoretical analysis, 

formal proofs, and comparative evaluations. The methods are structured to ensure a comprehensive 

understanding of the subject while contributing novel insights to the field of fuzzy topology. 

1. Literature Review and Conceptual Framework Development 

o A detailed review of existing literature was conducted to establish the foundational 

concepts related to fuzzy sets, fuzzy topologies, and fuzzy separation axioms. This review helped to 

identify gaps in current knowledge and guided the formulation of the research objectives. 

o Based on the literature review, a conceptual framework for fuzzy double s-Hausdorff 

spaces was developed. This framework includes the definition of fuzzy double s-Hausdorff spaces, the 

identification of key properties, and the establishment of their relationships with existing fuzzy 

topological spaces. 

2. Formal Definitions and Characterizations 

o The study begins by formally defining fuzzy double s-Hausdorff spaces within the 

context of fuzzy topology. Definitions are constructed to extend the classical Hausdorff condition into 

a fuzzy framework, taking into account the dual nature of the topologies involved. 

o Key characterizations of fuzzy double s-Hausdorff spaces are provided, highlighting 

their unique properties and distinguishing them from other related concepts such as fuzzy s-Hausdorff 

spaces and fuzzy Hausdorff spaces. 

3. Theorem Development and Proofs 

o A series of theorems are developed to establish the theoretical underpinnings of fuzzy 

double s-Hausdorff spaces. These theorems are carefully constructed to explore the implications of the 

fuzzy double s-Hausdorff condition in various topological scenarios. 

o Each theorem is accompanied by a formal proof, which rigorously demonstrates the 

validity of the proposed concepts. The proofs rely on established techniques in fuzzy topology, 

including set-theoretic approaches and logical reasoning within the fuzzy context. 

2.Preliminary Descriptions 

Definition:2.1 

Assume,  𝑋   remain a non-empty set. A FDS f remains an ordered pair (f1, f2) ∈ IX × IX suchthat  

f1 ≤ f2, where I =  [0,1] 

The family of FDS is denoted by FD( X ) 

Definition: 2.2 

A fuzzy topological space (X, τ) is supposed towards be fuzzy S- T2,  if for any twosome off different 

fuzzy points xt ,yu
 in  X, ∃ f, g ∈  τ  suchthat xt ∈ f , y

u
∈ g and  f g = 𝟎. 

 

 

 

 



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3. Fuzzy Double s- T2 

Definition: 3.1 

A FDTS( X , τ ) is supposed towards remain FDs-T2, meant for all two some of different FD points 

x (r,s), y
 (u,v)

 in 𝑋, near occurs binary fuzzy double open setsf = (f1, f2), g = (g1, g2) ∈ τ suchthat 

x (r,s) ∈  f, y (u,v) ∈ g that is  

f1(x) ≥ r, f2(x) ≥ s, g
1
(y) ≥ u, g

2
(y) ≥ v  and f ∩ g = 0. 

Definition: 3.2 

Assume,( X,  τ ) be a FDTS.  

Assume,X1 ⊆ X.  

Assume,f = (f1, f2) ∈ τ.  

Describe,f /X1 =  (f1/ X1, f2/ X1 ) 

suchthat ( f1 / X1 ) ( z ) =  f1( z ), ( f2 / X1 )( z ) =  f2(  z ) for all z ∈ X 1 

Define (τ / X1 ) = { ( f / X1 )  |f ∈ τ} 

Then (τ/X1) is called FD Subspace Topology on X1,then(X1, τ/X1) is said to beFD Subspace of 

( 𝐗, 𝛕 ) 

Proposition: 3.3 

sub-space of fuzzy double s-T2space is fuzzy double s-T2. 

Proof:  

Assume,(𝑋 , τ)remain a fuzzy double s-T2space.Assume,X1remain a sub-space of 𝑋. 

To-show:(X1, (τ /X1)) is a fuzzy double s-T2space  

consider,y (r,s), z(u,v) ϵ X1 such that y
(r,s)

≠ z(u,v)then y
(r,s)

, z(u,v) ∈ X, there exists Two Fuzzy Double 

Open Sets. 

f = (f1, f2), g = (g
1
, g

2
) ∈ τ suchthat  

f1 ( x ) ≥ r, f2( x ) ≥ s, g
1 

(y) ≥ u, g
2
(y) ≥ v also f  ⋂  g = 0 

Subsequently,X1 is a sub-space of 𝑋,𝑓 / X1, 𝑔 / X1 ∈ 𝜏 /  X1, where 

𝑓 / X1 = ( 𝑓1 / X1,  𝑓2 / X1 ), 𝑔 / X1 = ( 𝑔1/ X1, 𝑔2/ X1 ) 

∴ (f1 / X 1) ( y ) =  f1( y ) ≥ r 

(f2 / X 1) ( y ) =  f 2( y ) ≥ s 

(g
1
/  X1)(z)   =  g

1
(z) ≥ u 

(g
2
/ X1)(z)  = g

2
(z) ≥ v 

consider, 

( f / X1) ⋂ (g / X1 ) = (( f 1 / X1 ) ∧ (g
 1

 / X1 ), (f 2 / X1 ) ∧ (g
 2

 / X1 )) 



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((f1/ X 1 ) ∧ (g
1
/ X 1 )) ( y ) = (f1 / X 1 )( y )  ∧  (g

1
/ X 1 )(y) , for ally (r,s)  ∈  X1 ⊆ X 

= f1(y)∧g
1
( y ) , ∀ y

(r,s)
∈ X1 ⊆ X 

=(f1∧g
1
)( y ) , ∀ y (r,s) ∈ X1 ⊆  𝑋 

= 01(  y ), ∀ y
 (r,s) ∈ X1 ⊆  X 

( f1 / X1 ) ∧ (g
1
 / X1 ) =  01 

(( f 2 / X 1) ∧ (g
 2

 / X 1))( y ) = (f 2 / X 1)( y ) ∧ ( g
 2

 / X 1 )( y ) , for ally
 (r,s)

∈ X1 ⊆ X 

= f2(y) g
2
(y) , for ally

(r,s)
∈ X1 ⊆ X 

 = (f2 ∧ g
2
)(y) , for ally

(r,s)
∈ X1 ⊆ X 

= 02(y), for ally
(r,s)

∈ X1 ⊆ X 

=( f2 / X1 ) ∧ ( g
2
 / X1 ) = 02 

⇒(  f  /  X1)⋂ (  g  / X1 ) =(01, 02)= 0 

∴sub-space of FDs-T2space is aFDs-T2.  

Definition: 3.4 

Let f = (f1, f2) and g = (g
1
, g

2
) be binary FDS on X & Y correspondingly. Then cartesian product 

of f and g is a FDS in X xY definite as  

(f ∗ g)and is definite as f ∗ g = (f1 ∗ g
1
, f2 ∗ g

2
) where,   

( f
1

∗ g
1
)(x, y) = min{   f1( x ), g

1
( y )} and  

(  f
2

∗ g
2
 )  (x, y) = min{ f2( x ), g

2
( y )}, for every ( x, y) ∈ X × Y 

Definition: 3.5 

Assume,(𝑋, δ1) and (𝑌, δ2)remainBinary FDTS. At that point Product FDTδ1 × δ2 on IX ×

IYremains FDT taking the group {(  f ∗ g)| f = ( f1, f2) ∈ δ1, g =  ( g
1
, g

2
) ∈ δ2} as a basis. 

Proposition: 3.6 

product of binary FDs-T2is a FDs-T2. 

Proof:   

Assume,(𝑋, τ1) , (𝑌, τ2)remainbinary FDs-T2.  

To show: (𝑋 × 𝑌, τ1 × τ2)remainsFDs-T2 

Consider binary different Fuzzy Double-Points y
 (r,s)

, z(u,v) ∈ X × Y,where y = ( y
1
, y

2
) and  

z = ( z1, z2). Either  y
1

≠ z1 or  y
2

≠ z2 . 

Assume y
1

≠ z1 , so there exists Two Fuzzy Double Open Sets  



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 f = (f1, f2), g = (g
1
, g

2
)  suchthat 

f1(y
1
) ≥ r, f2(y

1
) ≥ s, g

1
(z1) ≥ u, g

2
(z1) ≥ v& 

 f  ⋂  g =  0 , everywhere 0 is a FD void set in X. 

 f ∗ 1 ∈ τ1 × τ2, subsequentlyf ∈ τ1 ,  1 ∈  τ2 and 

 g ∗ 1 ∈ τ1 × τ2, since  g ∈ τ1 ,  1 ∈  τ2, where 

 g ∗ 1 = (f1 ∗ 11, f2 ∗ 12) andg ∗ 1 = (g
1

∗ 11, g
2

∗ 12) 

Consider 

  (f1 ∗ 11)(y
1
, z1) = min  { f1(y

1
), 11(z1)} ≥ r 

  (f2 ∗ 12)(y
1
, z1) = min {f2(y

1
), 12(z1)} ≥ s 

  (g
1

∗ 11)(y
2
, z2) = min  { g

1
(y

2
), 11(z2)} ≥ u 

  (g
2

∗ 12)(y
2
, z2) = min { g

2
(y

2
), 12(z2)} ≥ v 

Also 

f  ∩  g = 0 

⇒( f1 ∧ g
1
, f2 ∧ g

2 
) = 0 

⇒ (f1 ∧ g
1
)(x) = 01(x) and (f2 ∧ g

2
)(x) = 02( x ),   𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 x ∈  X 

⇒ f1(x ) ∧ g
1
(x ) = 01(x ),   f2(x ) ∧ g

2
(x ) = 02(x ), 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦   x ∈ X 

⇒Also,f 1(x) = 01(x) or g
1
(x) = 01(x)& 

either f2(x) = 02(x)  or g
2
(x) = 02(x), ∀ x ∈ X 

⇒used for a FD universal set  1  in 𝑌 

moreover f1( x ) ∧  11( y ) =  0 or g
1
(x ) ∧ 11(y ) =  0, 

also,f2(x ) ∧  12(y ) =  0 or g
2
(x) ∧  12( y ) = 0, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 x ∈ X & y ∈ Y 

⇒moreover, (f1 ∗ 11)(x, y ) = 0 or  ( g
1

∗ 11)( x, y ) = 0 & 

Also, (f2 ∗ 12)( x, y ) = 0 or (g
2

∗ 12)( x, y ) = 0, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦  x ∈ X and   y ∈  Y 

⇒  ( f1 ∗ 11) ∧ ( g
1

∗ 11)( x, y ) = 0and 

( f2 ∗ 12) ∧ (  g
2

∗ 12)( x, y) = 0  , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦( x, y) ∈ X × Y 

⇒  ( f  ∗ 1 ) ∩ (g  ∗ 1) = 0  , where 0 ∈  X × Y. 

∴product of binary FDs-T2 is a FDs-T2. 

Definition: 3.7 

Assume,{(Xλ, δλ) |  λ ∈ Λ}remain a family of FDTS and X =
∏

λ ∈ Λ
Xλ 



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Let{fλ = (( f1) λ , ( f2) λ)) | λ ∈ Λ} and fλ is aFDS in Xλ. At that point their product 
∏

λ ∈ Λ
fλ is a FDS 

in 
∏

λ ∈ Λ
fλwell-defined as  

∏
λ ∈ Λ

fλ = (
∏

λ ∈ Λ
(f1)λ ,

∏
λ ∈ Λ

(f2)λ)  where  

∏
λ ∈ Λ

(f1) λ(x) = min{(f1)λ(xλ)}, ∀ x ∈ ∏ xλ ∈ Λ and  

∏
λ ∈ Λ

(f2)λ(x) = min{(f2)λ(xλ)}, ∀x ∈ ∏ xλ ∈ Λ 

The product topology on X is of the form 
∏

λ ∈ Λ
fλis a basis of FDOS, wherever 

fλ ∈ δλand fλ =  1 excludingmeant for finitely severalλ′s 

Proposition: 3.8 

 Arbitrary product of binary FDs-T2 is a FDs-T2. 

Proof:   

Assume,{(Xλ, (τ )λ) | λ ϵ Λ}remain a collection of FDs-T2. 

Assume,X = ∏ Xλ λ ϵ Λ    andτ = ∏ (τ ) λ λ ϵ Λ . 

 Consider two distinct fuzzy double points(xλ)(r,s), (y
λ
)

(u,v)
∈∏ Xλλ ϵ Λ , ∀λ ϵ Λ. 

 Therefore  (xμ)
(r,s)

≠ (y
μ
)

(u,v)
 for some μ ∈ Λ. Consequently, there exists Two Fuzzy Double Open 

Sets, 

(f)
 μ

= (( f1) μ, (f2) μ) and (g)
 μ

=  (( g
1
)

 μ
, ( g

2
)

 μ
) ∈ ( τ )μsuch that 

(𝑓1) μ(xμ) ≥ r ,( 𝑓2)μ(xμ) ≥ s, (𝑔1)μ(xμ) ≥ u ,(𝑔2)μ(xμ) ≥ vand   

(f)
μ

∩ (𝑔)
μ

= (0)
μ
 

   Let f = ∏ (f)
 λ  λ ϵ Λ , where ( f )

 λ
= (1)

λ
for 𝜆 ≠ μ and 

g  =  ∏  (g)
 λ

 λ ϵ Λ , where ( g )
 λ

= ( 1 )
  λ

 for 𝜆 ≠ μ.  

Then f, g ∈ ∏  (τ )  λ λ ϵ Λ  

f =  ∏ ( f )
 λ   λ ϵ Λ = (∏ ( f1)  λ  λ  ϵ Λ  , ∏ ( f2) λ λ  ϵ Λ ))and  

g  = ∏ ( g )
λ

λ ϵ Λ

= ( ∏ (g
1
)

λ
 λ  ϵ Λ

 , ∏(g
2
)

λ
λ  ϵ Λ

) 

∏ (f1) λ λ ϵ Λ (xλ) = min{( f1) λ(xλ)} , ∀𝜆∈ Λ 

                                =(  f1 )μ(xμ) ≥ r , for some μ ∈ Λ 



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∏(f1)λ

λ ϵ Λ

(xλ) ≥ r 

∏ (f2) λ λ ϵ Λ (xλ) = min{(f2) λ(xλ)} , ∀ 𝜆∈ Λ 

                                =(f2)μ(xμ) ≥ s , for some μ ∈ Λ 

∏(f2)λ

λ ϵ Λ

(xλ) ≥ s 

∏ (g
1
)

 λ λ ϵ Λ (xλ) = min{( g
1
) λ(xλ)} , ∀  𝜆∈ Λ 

                                =(g
1
)

μ
(xμ) ≥ u , for some μ ∈ Λ 

∏(𝑔1)λ

λ ϵ Λ

(xλ) ≥ u 

∏ (g
2
) λ λ ϵ Λ (xλ) = min{(g

2
) λ(xλ)} , ∀ 𝜆∈ Λ 

                                =(g
2
)

μ
(xμ) ≥ v , for some μ ∈ Λ 

∏(g
2
)

λ
λ ϵ Λ

(xλ) ≥ v 

Consider   

∏ ( f )
 λ λ ϵ Λ  ∩  ∏ ( g )

λ
 λ ϵ Λ =( ∏ (f1) λ λ ϵ Λ , ∏ ( f2) λ λ ϵ  Λ )  ∩ ( ∏ ( g

1
)

 λ λ ϵ Λ  , ∏ ( g
2
)

λ
 λ ϵ Λ ) 

                                              =(∏ ( f1 )λ λ ∈  Λ ∧ ∏ ( g
1
)

λ
 λ  ∈ Λ  , ∏  ( f2)λ ∧ ∏ ( g

2
)

 λ λ ∈ Λ λ ∈ Λ ) 

Then 

(∏  (f1) λ λ ∈ Λ ∧ ∏ (g
1
)

 λ λ ∈ Λ )(xλ)=(∏ ( f1)λλ ∈ Λ (xλ)) ∧ (∏ ( g
1
) λ(xλ)λ ∈ Λ ), for every 𝜆∈ Λ 

                                                               =  (min{(f1)λ(xλ)})∧ (min{(g
1
)

λ
(xλ)}) ,∀𝜆∈ Λ 

                                                               = ( f1)μ(xμ)∧(g
1
)

μ
(xμ) 

                                                               =((f1)μ ∧ (g
1
)

μ
) (xμ) 

                                                               =0  

(∏ (f2) λ λ ∈ Λ ∧ ∏ (g
2
) λ λ ∈ Λ )(xλ)=(∏ ( f2) λ λ ∈ Λ (xλ)) ∧ (∏ (g

2
)

 λ
(xλ) λ ∈ Λ ), for every 𝜆∈ Λ 

                                                               =  (min{(f2)λ(xλ)})∧ (min{(g
2
)

λ
(xλ)}) ,∀𝜆∈ Λ 

                                                               = (f2)μ(xμ)∧(g
2
)

μ
(xμ) 

                                                               =((f2)μ ∧ (g
2
)

μ
) (xμ) 

                                                               =0  



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∏ (f)
 λ

   λ ϵ Λ

∩ ∏ (g)
 λ

 λ ϵ Λ

= 0 

Consequently, Arbitrary Product of Two FDs-T2 is a FDs-T2. 

Results:  

 The study on fuzzy double s-Hausdorff spaces has yielded several significant findings, 

contributing to both the theoretical framework of fuzzy topology and its potential applications. The 

results are presented below, focusing on the key aspects of definitions, theorems, properties, and 

comparative analyses. 

1. Definition and Characterization of Fuzzy Double s-Hausdorff Spaces 

• Formal Definition: The study successfully formulated the definition of fuzzy double s-

Hausdorff spaces. These spaces are characterized by two fuzzy topologies on the same underlying set, 

where each topology satisfies a fuzzy version of the Hausdorff condition. Specifically, the separation 

of points is controlled by fuzzy parameters that account for the degree of uncertainty in the space. 

• Characterization Theorems: Several theorems were established to characterize the properties 

of fuzzy double s-Hausdorff spaces. These theorems provide the necessary and sufficient conditions 

for a fuzzy topological space to be considered a fuzzy double s-Hausdorff space. The results 

demonstrate that the spaces extend the classical concept of Hausdorff spaces by allowing for a graded 

separation between points. 

2. Properties of Fuzzy Double s-Hausdorff Spaces 

• Separation Properties: The study found that fuzzy double s-Hausdorff spaces exhibit unique 

separation properties that distinguish them from classical Hausdorff spaces and other fuzzy topologies. 

Specifically, the spaces allow for the separation of points to be controlled by two distinct fuzzy 

parameters, offering greater flexibility in handling scenarios where points are not strictly separable.  

• Intersection and Union Behavior: The results also showed that the behavior of fuzzy double 

s-Hausdorff spaces under intersection and union operations is consistent with the expected behavior 

of fuzzy topological spaces. However, the dual nature of the topologies introduces additional 

complexity, making these operations more intricate than in single fuzzy topology spaces 

Conclusions: 

 This study provides a comprehensive analysis of fuzzy double s-Hausdorff spaces, offering 

significant theoretical contributions and highlighting their potential applications. While there are areas 

for further exploration, the findings presented here lay the groundwork for future research and 

underscore the importance of fuzzy concepts in modern topology. 

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