The most three operations that has a special significance in fuzzy set theory are t.

1) union 2) intersection and, 3) complement.

These can be generalized to fuzzy sets in many ways.

Suppose X be a non empty set, a class τ which is a subset of X is defined a topology if it satisfies the following conditions:

(i) X and φ belong to τ .

(ii) The union and intersection of any number of sets in τ belongs to τ .

The pair ( X , τ ) is called a topological space and the number of τ are described as τ -open sets in X.

Example:

For any subsets of X = { 1 , 2 , 3 , 4 , 5 } .

τ 1 = { X , φ , { 1 } , { 3 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 , 5 } } τ 2 = { X , φ , { 1 } , { 3 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 } } τ 3 = { X , φ , { 1 } , { 3 , 4 } , { 1 , 3 , 4 } , { 1 , 2 , 4 , 5 } }

We noticed that τ 1 is a topology on X but τ 2 and τ 3 is not a topology on X. Since τ 1 satisfies the necessary two conditions (i), and (ii) where as the union { 1 , 3 , 4 } { 2 , 3 , 4 } = { 1 , 2 , 3 , 4 } does not belong to τ 2 and the intersection { 1 , 3 , 4 } { 1 , 2 , 4 , 5 } = { 1 , 4 } of two sets in τ 3 does not belong to τ 3 i.e. τ 2 and τ 3 does not satisfy the condition (ii).

Discrete topology

Assume X be a nonempty set. Then the group of all subsets including the empty set, of X, known as Power set P ( X ) is a topology on X and is define as discrete topology.

Example:

If X = { p , q } , and τ = { X , { p } , { q } } .

Then τ is discrete topology on X.

Indiscrete topology

For any non empty set X the collection of set consisting ϕ and X is a topology on X, is defined as indiscrete topology.

Example:

Let, X = { p , q , r } then τ = { X , ϕ } is a topology on X and is known as indiscrete topology.

For a open set x ∈ E ⊆ B a point x ∈ B is an interior point iff x ∈ E ⊆ B which is denoted by x ∈ E ⊆ B int(B) where B is a subset of topological space. The interior of B C is known as the exterior of B and is denoted by ext. The point which is interior nor an exterior of B is known as boundary point of B.

i.e. bd(B) = Complement of (int(B) ∪ ext(B)).

The intersection of all closed super sets of B is denoted by B ¯ or B − is known as closure of B.

Where, ( X , τ ) is the topological space and B is the subset of it. If { E i : i ∈ I } , denote the class of all closed subsets of X containing B, then B ¯ will be the closure of a set B, if

(i) B ¯ is closed and B = B ¯ .

(ii) if E is a closed super set of B, then B ⊂ B ¯ ⊂ E .

Properties of closure operator

For any topological space ( X , τ ) if P and Q be two arbitrary subsets of X, then the operation of forming closures hold the following four properties:

(i) φ ¯ = φ ;

(ii) P ⊆ P ¯ ;

(iii) P ¯ ¯ = P ¯ ;

(iv) P ∪ Q ¯ = P ¯ ∪ Q ¯ .

A continuous one–to-one mapping of a topological space onto another is defined to be homomorphic if there exist a homeomorphism of X onto Y and their points can be set into one-to-one correspondence in such a manner that their open sets also correspond to one another.

In the past, the leading attempt to build up the fuzzy counterpart of basic topology was commenced by Chang [7] in 1968. From the point of Chang, on a set X a fuzzy topology is a family F = { μ : μ isfuzzysetin X } of fuzzy subsets (i.e. F ∈ I X ) that satisfies the following three axioms:

(i) 0 , 1 ∈ F ;

(ii) μ 1 , μ 2 ∈ F , then μ 1 ∧ μ 2 ∈ F ;

(iii) If { μ i : i ∈ j } ⊂ F , where j denotes an index set, then ∨ J   μ i ∈ F ;

F is described as a fuzzy topology for X and the pair ( X , F ) is named as a fuzzy topological space or in short f.t.s. The members of F are defined as F-open fuzzy set. If the complement of ρ , denoted by ρ C , is F-open then an element ρ ∈ [ 0 , 1 ] X is said to be a closed fuzzy set.

Indiscrete Fuzzy topology

The same as in general fuzzy topology, the indiscrete fuzzy topology contains only fuzzy sets 0 and 1.

Example 1:

Indiscrete fuzzy topological ( X , F ) , F = { ρ ∈ I X : α constantfuzzyset }

i.e. for all, ρ ∈ F , if ρ = constant

Then, ρ ( x ) = 0 ,   ∀ x

ρ ( x ) = 1 ,   ∀ x

Wherever, I X is the fuzzy set on X.

Discrete fuzzy topology

Discrete fuzzy topology is the set that included all the fuzzy sets.

Example 2:

X = { x , y , z } , α = ( 0.8 , 0.9 , 0.7 ) , β = ( 0.6 , 0.5 , 0.4 ) , δ = ( 0.3 , 0.3 , 0.2 )     and     F = { 0 , α , β , δ , 1 } .

Now,

α ∧ β = ( 0.8 , 0.9 , 0.7 ) ∧ ( 0.6 , 0.5 , 0.4 ) = ( 0.6 , 0.5 , 0.4 ) = β ∈ F α ∧ δ = ( 0.8 , 0.9 , 0.7 ) ∧ ( 0.3 , 0.3 , 0.2 ) = ( 0.3 , 0.3 , 0.2 ) = δ ∈ F β ∧ δ = ( 0.6 , 0.5 , 0.4 ) ∧ ( 0.3 , 0.3 , 0.2 ) = ( 0.3 , 0.3 , 0.2 ) = δ ∈ F

And α ∨ β = ( 0.8 , 0.9 , 0.7 ) ∨ ( 0.6 , 0.5 , 0.4 ) = ( 0.8 , 0.9 , 0.7 ) = α ∈ F α ∨ δ = ( 0.8 , 0.9 , 0.7 ) ∨ ( 0.3 , 0.3 , 0.2 ) = ( 0.8 , 0.9 , 0.7 ) = α ∈ F β ∨ δ = ( 0.6 , 0.5 , 0.4 ) ∨ ( 0.3 , 0.3 , 0.2 ) = ( 0.6 , 0.5 , 0.4 ) = β ∈ F

Also

α ∨ β ∨ δ = ( 0.8 , 0.9 , 0.7 ) ∨ ( 0.6 , 0.5 , 0.4 ) ∨ ( 0.3 , 0.3 , 0.2 ) = ( 0.8 , 0.9 , 0.7 ) = α ∈ F

It is obvious 0 , 1 ∈ F .

Therefore, ( X , F ) is a fuzzy topological space.

Example 3:

Consider X be an abstract set equipped with the family F 1 of fuzzy sets, where F 1 = { υ : υ ( x ) ≥ 1 2 , ∀ x ∈ X } ∪ { α : α constant < 1 2 }

Then from the properties of fuzzy topological space

(i) 0 , 1 ∈ F 1 ;

(ii) if ρ 1 , ρ 2 ∈ F 1 ; then ρ 1 ( x ) ∧ ρ 1 ( x ) ⇒ ( ρ 1 ∧ ρ 2 ) ( x ) ≥ 1 2 ⇒ ( ρ 1 ∧ ρ 2 ) ∈ F 1

i.e. finite intersection belongs to F 1 .

(iii) ρ i ∈ F , ∀ i ∈ j , where j denotes an index set.

so, ∨ i ∈ j   ρ i ≥ 1 2 ∈ F 1

Hence, we conclude that the pair ( X , F ) is a topological space.

For any μ the closure of fuzzy topological space is denoted by μ ¯ and is defined as the smallest closed fuzzy set that containing μ .

Equivalent μ ¯ is defined as following way:

μ ¯ = { α : α is F -closedand α ≥ μ }

Obviously, then μ ¯ is always F-close.

Now we have discussed and proved some properties of Fuzzy closure operators.

A map μ → μ ¯ from [ 0 , 1 ] X into [ 0 , 1 ] X is said to be a closure operation if for all μ , λ ∈ [ 0 , 1 ] X it satisfies the four list of properties:

(i) μ ≤ μ ¯ ;

(ii) μ ¯ ¯ ≤ μ ¯ (i.e. the closure operation is idempotent);

(iii) μ ∨ λ ¯ = μ ¯ ∨ λ ¯ ;

(iv) 0 ¯ = 0 ;

Proof (i):

From the definition of closure operator we know,

μ ¯ = ∧ { υ : υ closed& υ ≥ μ } .

Thus, μ ≤ μ ¯ .

Proof (ii):

Since μ ¯ ¯ is the smallest closed set containing μ ¯ and μ ¯ itself is closed,

Then, μ ¯ ¯ = μ ¯ .

Proof (iii):

Clearly α ¯ ∨ β ¯ is closed.

Again,

α ¯ ∨ β ¯ ≥ α ∨ β ⇒ α ¯ ∨ β ¯ ¯ ≥ α ∨ β ¯ ⇒ α ¯ ∨ β ¯ ≥ α ∨ β ¯ (1)

[Since α ¯ ∨ β ¯ is closed, so α ¯ ∨ β ¯ ¯ = α ¯ ∨ β ¯ ]

Again,

α ∨ β ≥ α ⇒ α ∨ β ¯ ≥ α ¯ (a)

Similarly, α ∨ β ¯ ≥ β ¯ (b)

From (a) and (b) we have,

α ∨ β ¯ ≥ α ¯ ∨ β ¯ (2)

So from (1) and (2), we have,

α ¯ ∨ β ¯ = α ∨ β ¯ .

Proof (iv):

Since the whole space 1 ∈ F is open, then its complement i.e. 1 C = 0 , is closed

Also, 0 ¯ is closed.

So, we can write 0 ¯ = 0 .

Definition. Fuzzy topology generated by closure operator is denoted by F X and is define by

F X = { μ ∈ [ 0 , 1 ] X : 1 − μ ¯ = 1 − μ } ,

Then ( X , F X ) is called closure of fuzzy topological space (f.t.s.) generated by closure operator.

Theorem 4.1

Suppose α , β are fuzzy sets in X and ( X , F ) is a fuzzy topological space. Then prove that α ¯ ∧ β ¯ = α ∧ β ¯ .

Proof:

Clearly, α ¯ ∧ β ¯ is open. Again,

α ¯ ∧ β ¯ ≤ α ∧ β ⇒ α ¯ ∧ β ¯ ¯ ≤ α ∧ β ¯ ⇒ α ¯ ∧ β ¯ ≤ α ∧ β ¯ (3)

Again,

α ∧ β ≤ α ⇒ α ∧ β ¯ ≤ α ¯ (4)

Similarly, α ∧ β ¯ ≤ β ¯ (5)

From (4) and (5) we have,

α ∧ β ¯ ≤ α ¯ ∧ β ¯ (6)

From (3) and (6), we have,

α ¯ ∧ β ¯ = α ∧ β ¯ .

The smallest superior bound of all interior fuzzy sets of μ is called the interior of μ , and is denoted by μ ∘ .evidently, μ ∘ ∈ F X . So, μ ∘ is F-open.

Suppose ( X , F ) be a fuzzy topological space and I = [ 0 , 1 ] , and μ , λ ∈ [ 0 , 1 ] X where μ ≥ λ . Then λ is defined as an interior fuzzy set of μ iff for ρ ∈ F X such that μ ≥ ρ ≥ λ .

Theorem 4.2

Assume ( X , F ) be a fuzzy topological space. Then

(i) 0 ∘ = 0 , 1 ∘ = 1

(ii) μ ∘ ≤ μ

(iii) μ ∘ ∘ = μ ∘

(iv) ( μ ∧ υ ) ∘ = μ ∘ ∧ υ ∘

Proof (i):

Since, the interior of any set joining of all open subset contained in this set.

Now, the empty set 0 and the whole space 1 of an f.t.s. is open.

Thus 0 ∘ = 0 ,1 ∘ = 1 .

Proof (ii):

From the definition of interior of a set μ , the combination of all open subsets included in μ , denoted by μ ∘ , i.e.

μ ∘ = ∨ { υ open& υ ≤ μ }

Hence we conclude μ ∘ ≤ μ .

Proof (iii):

Since μ ∘ itself is open and μ ∘ ∘ is the greatest open set contained in μ ∘ .

So evidently, μ ∘ ∘ = μ ∘ (Proved).

Proof (iv):

Since ( μ ∧ λ ) ∘ ≤ μ ∘ and

( μ ∧ λ ) ∘ ≤ λ ∘

So,

( μ ∧ λ ) ∘ ≤ μ ∘ ∧ λ ∘ (7)

On the other hand,

μ ∘ ∧ λ ∘ ≤ μ ∧ λ , of which the open set μ ∘ ∧ λ ∘ hold in μ ∧ λ ;

Hence, μ ∘ ∧ λ ∘ must be contained in the largest open set ( μ ∧ λ ) ∘

i.e. μ ∘ ∧ λ ∘ ≤ ( μ ∧ λ ) ∘ (8)

From (7) and (8) we conclude that

( μ ∧ λ ) ∘ = μ ∘ ∧ λ ∘ (Proved).

Theorem 4.3

Assume α and β be two fuzzy sets in an f.t.s.. Then α ∘ ∨ β ∘ = ( α ∨ β ) ∘ .

Proof:

α ∘ is open and β ∘ is open .

So α ∘ = α , and β ∘ = β .

Therefore α ∘ ∨ β ∘ = ( α ∨ β )

Then, ( α ∘ ∨ β ∘ ) ∘ = ( α ∨ β ) ∘

Again, α ∘ ∨ β ∘ is open, so ( α ∘ ∨ β ∘ ) ∘ = α ∘ ∨ β ∘

Therefore, α ∘ ∨ β ∘ = ( α ∨ β ) ∘

Hence, α ∘ ∨ β ∘ = ( α ∨ β ) ∘ (Proved).

For μ ∈ [ 0 , 1 ] X , the fuzzy boundary is denoted by μ b is define to be the minimum of all F-closed sets ρ with the property ρ ( x ) ≥ μ ¯ ( x ) , ∀ x ∈ X for which we have [ μ ¯ ∧ ( 1 − μ ) ] ( x ) > 0 .

Obviously, μ b is F-closed and μ b ≤ μ ¯

Fuzzy point

For any fuzzy set, μ β ∈ [ 0 , 1 ] X , fuzzy point is described as,

μ β ( x ) = { β if   x = x ∘ 0           otherwise

x ∘ is the support of the fuzzy point μ β .

Suppose ( X , F ) be a fuzzy topological space, if μ , λ ∈ [ 0 , 1 ] X and μ ≥ λ then μ is said to be a fuzzy neighborhood (in short n.b.d) of λ if there is a γ ∈ F X so that μ ≥ γ ≥ λ . Example:

We know if ( X , F ) is a fuzzy topological space, and α , β ∈ F , then β is called fuzzy n.b.d. of α if and only if α ≤ β .

Now, let

X = { u , v , w } ,   F = { 0 , 1 , α , β , γ } ,   α = ( 0.5 , 0.3 , 0.2 ) , β = ( 0.7 , 0.1 , 0 ) ,   γ = ( 0.5 , 0.1 , 0 ) .

F 1 α ≡ ( 0.5 , 0.1 , 0.1 ) ≤ α F 2 α ≡ ( 0.0 , 0.3 , 0.0 ) ≤ α F 3 α ≡ ( 0.0 , 0.0 , 0.2 ) ≤ α

where F 1 α , F 2 α and F 3 α denote fuzzy points.

Here, F 1 α ∨ F 2 α ∨ F 3 α = α , but F 1 α ∨ F 2 α ∨ F 3 α ≠ β .

Also since,

F 1 α ≤ β F 2 α ≤ β F 3 α ≤ β

So, we conclude that α ≤ β .

Therefore β is a fuzzy n.b.d. of α .

Suppose ( X , F 1 ) and ( Y , F 2 ) be two fuzzy topological space and let F : ( X , F 1 ) → ( Y , F 2 ) be a function from X into F. Then F is said to be continuous at a point x ∈ X if the inverse image F − 1 ( υ ) of all F 2 open subset υ of Y is a F such that F − 1 ( υ ) ∈ F 1 . Figure 6 displays the continuous open fuzzy set.

Theorem 4.5

Let ( X , F 1 ) and ( Y , F 2 ) be two fuzzy topological spaces. Then a function F : ( X , F 1 ) → ( Y , F 2 )

is continuous if

F ( μ ¯ ) ≤ F ( μ ) ¯ for all μ ∈ I Y .

Proof:

Let μ 1 ∈ I Y be such that μ 1 C ∈ F 2 and put μ = F − 1 ( μ 1 ) , then

F ( μ ¯ ) ≤ F ( μ ) ¯ = F ( F − 1 ( μ 1 ) ) ¯ ≤ μ 1 ¯ = μ 1 So, But μ ¯ is such that μ ≤ μ ¯ .

So, μ = μ ¯ and μ is closed,

I.e. F − 1 ( μ 1 ) is closed,

I.e. F is continuous

Conversely, let f is fuzzy continuous. We must show that for any μ ∈ I X . F ( μ ¯ ) ≤ F ( μ ) ¯

We know that

F ( μ ¯ ) = inf { υ ∈ I Y : F − 1 ( υ ) ≥ μ ¯ }

and F ( μ ) ¯ = inf { ξ ∈ I Y : ξ C ∈ F 2 , ξ ≥ F ( μ ) }

It suffices to prove that for all ξ ∈ I Y such that ξ C ∈ F 2 and ξ ≥ F ( μ ) , there exists υ ∈ I Y such that F − 1 ( υ ) ≥ μ ¯ and ξ ≥ υ .

Let, ξ ∈ I Y be such that ξ ≥ υ and ξ ≥ F ( μ ) .

Put υ = ξ , then ξ ≥ υ and f − 1 ( ξ ) ≥ F − 1 ( f ( μ ) ) ≥ μ and since ξ C ∈ F 2 and F is fuzzy continuous implies that F − 1 ( ξ ) ≥ μ ¯ .

Hence the theorem is proved.

Let F : ( X , F 1 ) → ( Y , F 2 ) be a function. Then

i) F is called continuous if ∀ υ ∈ F 2 , F − 1 ( υ ) ∈ F 1 .

ii) F is called open if ∀ μ ∈ F 1 , F ( μ ) ∈ F 2 .

iii) F is called closed if ∀ F 1 -closed set μ , F ( μ ) is F₂-closed.

iv) F is called homomorphism if F is bijective (i.e. one-one onto) and biconditional. It means that, both f and F − 1 are continuous.

Dense of fuzzy topological spaces

Let ( X , F ) be a fuzzy topological space. Some definitions on this space are given below:

Definition 1.

A fuzzy set μ is said to be fuzzy dense or everywhere dense if and only if μ ¯ = 1 .

Definition 2.

A fuzzy set μ is said to be fuzzy-nowhere dense if and only if ( μ ¯ ) C = 1 .

Example:

int ( μ ¯ ) = 0 , that is if the interior of closure of any fuzzy set μ is empty, then μ is called nowhere dense in X,

Definition 3.

Any fuzzy set μ is said to be fuzzy boundary (F-boundary) if and only if 1 − μ ¯ = 1 .

(i) If λ ≥ μ and λ is F-dense then μ is F-dense too.

Proof:

Since, λ ≥ μ

⇒ μ ≤ λ ¯ ⇒ μ ¯ ≤ λ ¯ ¯ ⇒ μ ¯ ≤ λ ¯ [ λ ¯ ¯ = λ ¯ ]

Since, λ ¯ = 1 , thus μ ¯ = 1 .

Thus μ is F-dense too.

(ii) If λ ≤ μ and μ is F-boundary, consequently then λ is F-boundary too.

Proof:

Since λ ≤ μ

⇒ 1 − λ ≥ 1 − μ ⇒ 1 − λ ¯ ≥ 1 − μ ¯

So, 1 − μ ¯ = 1 , (Since μ is F-boundary).

Thus, 1 − λ ¯ = 1 .

Thus, λ is F-boundary too

(iii) If μ is F-nowhere dense and μ ≥ λ , then λ is F-nowhere dense.

Proof:

Given, λ ≤ μ

λ ≤ μ ¯ ⇒ 1 − λ ≥ 1 − μ ¯ ⇒ 1 − λ ¯ ≥ 1 − μ ¯ ¯

Since μ is F-nowhere dense, which implies 1 − λ ¯ = 1 , from which we conclude that λ is F-nowhere dense.

(iv) If μ is F-nowhere dense, then so is μ ¯ .

Proof:

Since μ ≤ μ ¯

⇒ 1 − μ ≥ 1 − μ ¯ ⇒ 1 − μ ¯ ≥ 1 − μ ¯ ¯ ⇒ μ C ¯ ≥ ( μ ¯ ) C ¯ (9)

Since μ is nowhere dense, so μ C ¯ = 1 .

Hence from (9) we conclude that μ ¯ is F-nowhere dense.

(v) If μ ∈ [ 0 , 1 ] X is F-dense and λ ∈ F , then λ ≤ μ ∧ λ ¯ .

Proof:

We know from the properties of closure operation

μ ∧ λ ≤ μ ∧ λ ¯

Also, μ ∧ λ ¯ ≤ μ ¯ ∧ λ ¯

Thus,

μ ∧ λ ≤ μ ∧ λ ¯ ≤ μ ¯ ∧ λ ¯ (10)

Also,

μ ¯ ∧ λ ¯ ≤ λ ¯ (11)

And we know,

λ ¯ ≥ λ (12)

Hence, from (10), (11), and (12) we conclude that

μ ∧ λ ¯ ≥ λ .

Proposition 4.1

If μ is F-dense then, ( 1 − μ ) ∘ = 0 . That is

Proof:

We claim from the definition of interior fuzzy set,

( 1 − μ ) ∘ = 1 − μ ¯ .

Now,

1 − μ ¯ = 1 − ∧ { τ : ( 1 − p ) ∈ F , p ≥ μ } = ∨ { ( 1 − p ) ∈ F : υ ≥ τ } = ∨ { τ ∈ F : 1 − τ ≥ μ } = ∨ { τ ∈ F : 1 − μ ≥ τ } = ( 1 − μ ) ∘

But since μ is F-dense, so, μ ¯ ≡ 1 .

Hence, 1 − μ ¯ = 0 .

Base

Let ( X , F ) be a fuzzy topological space and P is a sub family of F, then P is called a base of F if and only if every member of F can be represent as supremum of member of P.

Example:

Let X = { x , y } , and α , β , γ ∈ I X , where α = ( 0.3 , 0.6 ) , β = ( 0.7 , 0.5 ) , γ = ( 0.7 , 0.6 ) and

( X , F ) be a fuzzy topological space, where F = { 0 , 1 , α , β , γ , δ } and B = { 0 , α , β , λ , 1 } .

Then, 0 , α , β , γ and 1 are in P.

Now, γ = ( 0.7 , 0.6 ) = α ∨ β

Hence P is a base for F.

Subbase

Suppose ( X , F ) is a fuzzy topological space and B is a non empty subset of F, then B is called a subase of F, if and only if the finite intersection of members of B forms a base for F.

Example:

Assume X = { x , y } and α , β , γ , δ ∈ I X where, α = ( 0.2 , 0.6 ) , β = ( 0.3 , 0.5 ) , γ = ( 0.2 , 0.5 ) and δ = ( 0.3 , 0.6 ) and δ = ( 0.3 , 0.6 ) .

Then ( X , F ) is a fuzzy topological space, where F = { 0 , 1 , α , β , γ , δ } and B = { α , β } is a subbase and P = { 0 , 1 , α , β , γ } is a base of F.

An fuzzy topological space ( X , F ) is said to be Hausdroff iff ∀ x , y ∈ X , x ≠ y , there exist μ , λ ∈ F so that μ ( x ) = λ ( y ) = 1 and μ ∧ λ = 0 .

Theorem 4.7

Let ( X , F ) is a Hausdroff fuzzy topological space, while ( Y , F 1 ) a fuzzy topological space and F : X → Y is an Fopen bijection, then ( Y , F 1 ) is also a Hausdroff fuzzy topological space.

Proof:

Let y 1 , y 2 ∈ Y and y 1 ≠ y 2 .

Let, y 1 = f ( x 1 ) and y 2 = f ( x 2 ) . Since by hypothesis ( X , F ) is a Hausdroff topological space, then there exist μ , λ ∈ F such that μ ( x 1 ) = λ ( x 2 ) = 1 and μ ∧ λ = 0 .

Now, f ( μ ) and f ( λ ) both belong to F₁, since f is an F-open map.

Also f ( μ ∧ λ ) = f ( μ ) ∧ f ( λ ) = F ( 0 ) = 0 .

Finally,

f ( μ ) ( y 2 ) = sup x ∈ f − 1 ( y 1 ) μ ( x ) = 1 and f ( λ ) ( y 1 ) = sup x ∈ f − 1 ( y 2 ) λ ( x ) = 1

Which satisfy all the condition of Hausdroff fuzzy topological space.

Hence, indeed ( Y , F 1 ) is a Hausdroff fuzzy topological space.