A theoretical approach on intuitionistic Fuzzy Hausdor ﬀ space

The object of this paper is to introduce a new de ﬁ nition for intuitionistic fuzzy Hausdor ﬀ space (IFHS). We investigate some of its characterizations and discuss it with some necessary counter examples. In addition, we compared the new notion with the existing notions. Finally we point out the signi ﬁ cance of Hausdor ﬀ ness in digital image processing


Introduction
The concept of fuzziness exists almost everywhere in our daily life.To confront the difficulty due to ambiguity, Zadeh [16] proposed the fuzzy theory in 1965 and it was generalised into notion of intuitionistic fuzzy sets (IFS) by Atanassov [1] in 1986.The research in fuzzy theory grew rapidly day by day and got its credit in all the branches of Mathematics.Fuzzy topological space (FTS) was studied by several authors like Chang [4] and Lowen [10].
The notion of intutionistic fuzzy topological space (IFTS) was introduced and studied by Coker [5], [6].Since every concept in topology is defined in terms of open sets, separation axioms especially T 2 axiom developed by Hausdorff plays a vital role in making non trivial and interesting statements.Though different versions of fuzzy Hausdorff spaces are available in literature, the notion of nearly fuzzy Hausdorff space (NFHS) developed by Ramakrishnan and Lakshmana Gomathi nayagam [14] was a generalised one.
A few definitions for IFHS were introduced and studied by several authors like Cooker [5], Gallego Lupianez [11], Lakshmana and Muralikrishnan [13] and A.k. Singh and R.Srivastava [15].Later on Md Sadadat Hossain [7] has given seven definitions for IF T 2 spaces out of which IF T 2 (iv) is the most generalised one.Saiful Islam [8] has also presented eight types of IF T 2 spaces and concluded that IF T 2 (viii) is the most generalised version.Recently Md.Aman Mahbub [2] worked on seperation axioms in intutionistic fuzzy compact topological spaces.
Though the definition [8] generalises all other existing definitions, it had some drawbacks.Let T = {a, b} and A = (M a .2b .8, N a .8b .1 ).Here A is an IFS which can be viewed according to [8], as a set to which a belongs and b does not belong, though the membership of a(M A (a) = 0.2) to lie in A is much lesser than the membership of b(M A (b) = 0.8) to lie in A. Also the membership of a to lie in A is much lesser than the non membership of a to lie in A. But while generalising the concept of "an element x to belong a crisp set A" to the concept of "an element belongs to a IFS A" it has to be logically assumed that the membership degree of x to lie in A is greater than the non membership degree of x to lie in A. So M A (a) > N A (a). Since M A (a) + N A (a) ≤ 1 we get N A (a) < 1  2 and M A (a) > 1 2 .Hence we can think of as a point a belongs to a IFS A if M A (a) > 1 2 .This concept [14] generalises the crisp concept while greater care is taken intuitively to meet the logical requirements for belongingness.This paper proposes a new definition of IFHS by adopting the concept [14], in order to rectify the above mentioned illogicality.Further we describe a definition for intuitionistic fuzzy closedness of a singleton and explore some of its characteristics.Some results in crisp topology spaces are discussed in intuitionistic fuzzy set up.We have also compared the proposed definition with the definitions available in literature.

Preliminaries
This section follows some elementary definitions.
where the functions M R : T → I and N R : T → I denote the degree of membership and the degree of non membership of t ∈ T to the set R respectively, and Let T be a nonempty set and the IFSs, R and S be in Definition 2.3.
[5] Let T be a universal set.We define 0 ∼ and 1 ∼ as follows : [4] An intuitionistic fuzzy topology (IFT) on a non empty set T is a family δ of IFSs in T satisfying the following axioms: The pair (T, δ) is called an IFTS and any IFS in δ is known as an intuitionistic fuzzy open set (IFOS).
Definition 2.5.[15] Let (α, β) ∈ (0, 1) and α + β ≤ 1.An intuitionistic fuzzy point Definition 2.6.[15] Let P t (α,β) be an IFP in T and R = ht, M R , N R i be an IFS in T .Then P t (α,β) is said to be properly contained in R, Definition 2.7.[9] Let (T, δ) be an IFTS on T and N be an IFS in T .Then N is said to be an ε-neighbourhood of an IFP P t (α,β) in T if there exist an IFOS, G in T such that Definition 2.14.[11] An IFTS (T, δ) called q−T 2 if for every distinct IFPS p, q in T , there exists ε-neighbourhood M and N of p and q respectively such that Definition 2.15.[13] An IFTS (T, δ) is said to be a nearly intuitionistic fuzzy Hausdorff space (NIFHS) if for every pair of elements x 6 = y of T , there exist non zero disjoint IFOS R and Definition 2.16.[8] An IFTS (T, δ) is said to be IFT2 if for every pair of elements x 6 = y of T , there exist two IFOS R and S of δ such that

Nearly intuitionistic fuzzy Hausdorff spaces
In this section a new notion of NIFHS is introduced and its properties are studied.This notion is compared with the existing notions.
Definition 3.1.Two IFSs R and S of T are said to intersect at Definition 3.2.Two IFSs R and S of T are said to be disjoint if they do not intersect at any point of T that is, N R (t) + N S (t) ≥ 1, for all t ∈ T .
Definition 3.3.An IFTS (T, δ) is said to be NIFHS if for every pair of elements x 6 = y of T , there exist non zero disjoint IFOSs R and Theorem 3.4.Let (T, δ) be a NIFHS.Then a subspace of a NIFHS is a NIFHS.

Proof.
Let (T, δ) be a NIFHS and Theorem 3.5.Let (T i , δ i ) i∈I be a family of NIFHS.Then arbitrary product of NIFHS is a NIFHS.

Proof.
Let I be an indexed set and (T i , δ i ) i∈I be a family of NIFHS.Let (T = ΠT i , δ = Πδ i ) be the PIFT in which each projection mapping forms sub-base for PIFT.To prove that (T, δ) is a NIFHS, consider x 6 = y ∈ ΠT i .So there exist atleast one j ∈ I such that x j 6 = y j .Since (T j , δ j ) is a NIFHS, there exist two open sets R j , S j such that M R j (x j ) > 1  2 , M S j (y j ) > 1  2 and N R j (t j ) + N S j (t j ) ≥ 1, for all t j ∈ T j .Clearly θ −1 j (R j ) and θ −1 j (S j ) are the members of σ and hence elements of δ.
This is a contradiction.Hence (T, δ) is a NIFHS. 2 Theorem 3.6.In a NIFHS (T, δ), any sequence of points of T converges intuitionistic fuzzily to unique point, if it converges.

Proof.
Assume that {t n } converges intuitionistic fuzzily to distinct points x and y.Since (T, δ) is a NIFHS, there exist two open sets R and , for all n ∈ Z + and so t n doesn't converge to t fuzzily).But it is not NIFHS.For let x 6 = y ∈ T .Suppose there exist R, S ∈ δ such that M R (x) > 1 2 , M S (y) > 1 2 , and N R (t) + N S (t) ≥ 1, for every t ∈ T .Since R, S ∈ δ, N R and N S have countable support {x n } n∈Z + and {y m } m∈Z + respectively.As T is uncountable, there exist t ∈ T − {x n , y m } n,m∈Z + such that N R (t) = 0, N S (t) = 0, which contradicts the fact that N R (t) + N S (t) ≥ 1, for every t ∈ T .

Proof.
Let y 1 , y 2 in Y be two distinct points.Since f is bijective, there exist unique distinct points Remark 3.10.The requirement that f is IF open can not be dropped in the above theorem.
Remark 3.12.The requirement that f is onto can not be dropped in the above theorem.
Remark 3.14.The requirement that f is one-one can not be dropped.

Proof.
To prove (Y, σ) is NIFHS, take two distinct points y 1 , y 2 ∈ Y .Since f is bijective, there exist unique t Note.Nearly intuitionistic fuzzy Housdorffness is a topological property.

Intuitionistic fuzzy Hausdorff spaces
Here singletons are IF closed as per (ii) but not by (i).
. Clearly (T, δ) is NIFHS.Also note that every singletons are IF closed by above two definitions.

Note.
The above examples shows that the second definition is the generalised one.That is singleton {x} is IF closed as per (i) ⇒ (ii) but not the converse.
Note.The following results based on the generalised definition.
Note.In NIFHS, singletons need not be IF closed.This can be proved by the following example.
Definition 4.6.An IFTS (T, δ) is said to be a IFHS if (T, δ) is NIFHS and every singletons are IF closed.

Proof.
Let (T i , δ i ), i = 1, 2, ..., n be IFHS.By theorem (3.5), (T = ΠT i , δ = Πδ i ) is NIFHS.Now, we prove that singletons are IF closed in PIFHS.Let t = (t 1 , t 2 , ..., t n ) ∈ T be arbitrary.Since t i ∈ T i , and (T i , δ i ) is IFHS, there exist IFOS A i ∈ δ i such that M A i (t i ) < 1 2 and M A i (y i ) = 1, for all y i 6 = t i ∈ T i .We know that the projection, θ i : ΠT i → T i is IF continuous in PIFT.As A i ∈ δ i , for every i, {θ −1 i (A i )/A i ∈ δ i , i = 1, 2, ..., n} is the collection of sub-base.Now Remark 4.9.Arbitrary product of IFHS need not be IFHS as can be seen in the following example.
Suppose for x ∈ T , {x} is closed then there should exit some IFOS A ∈ ( Since A ∈ ( Q δ α ), A can be written as arbitrary union of basic elements.
which implies W M A α (x) < 1 2 and sup α M A α (y) = 1, for all y 6 = x.For given ε > 0 there should exist A α such that M A α (x) < 1 2 and M A α (y) > 1 − ε, for all y 6 = x.Also A α is the finite intersection of subbasis elements.That is A , for all y 6 = x with θ i (x) = x i ,θ j (x) = y j and also for θ , for all y 6 = x with θ i (x) = x i , θ j (x) = y j and also for θ 2 , for all y 6 = x with θ i (x) = x i ,θ j (x) = y j and also for θ i (x) = y i ,θ j (x) = x j .Therefore θ −1 i (A 4i ) / ∈ C, for all i.So C is empty.That is there is no A α satisfying the requirement that M Aα (x) < 1 2 and M Aα (y) > 1 − ε, ∀y 6 = x, which is a contradiction.Therfore {x} is not closed.Hence (T, δ) is not IFHS.
Note.If we change our definition as "singleton {x} ⊆ T is said to be IF closed, if there exist an IFOS O with M O (x) ≤ 1 2 and M O (z) = 1, for all z ∈ T where z 6 = x"; then arbitrary product of IFHS is IFHS.

Comparative studies
The following study shows that the class of all IFTS introduced in this paper is finer than the classes in [5], [11] and also coarser than the classes Consider a IFTS (T, δ) in which H c holds.Then for every pair of distinct points x, y ∈ T there exist disjoint IFOSs A and B of δ such that M A (x) > 1 2 , M B (y) > 1  2 .This will follow immediately such that there exist two open sets A and B such that M A (x) > 0, N A (x) < 1, M A (y) < 1, N A (y) > 0, and M B (y) > 0, N B (y) < 1, M B (x) < 1, N B (x) > 0. Now we have to prove the condition (M A ∩ M B ) ⊂ (N A ∪ N B ).We have N A (t) ≥ M B (t) and N B (t) ≥ M A (t) for every t ∈ T .This will imply N A (t) ∪ N B (t) ⊇ M B (t) ∪ M A (t) ⊇ M B (t) ∩ M A (t). Hence H d holds., q f (.7,.1) there does not exists ε -neighbourhood M and N of p and q respectively.Example 5.6.H b 6 ⇒ H a Let T = {e, f }, δ = {(0, 1), (1, 0), (M e .6 f 0 , N e .3f 1 ), (M e 0 f .6 , N e 1 f .3 ), (M e .6 f .6 , N e .3f .3 ), }.Clearly (T, δ) is IFTS in which H b holds.But H a does not holds.

Hausdorffness in image processing
A raster image is a 2D array of numbers representing pixel intensities.In image processing points are analogous to pixels.A feature vector is a vector of numbers where each number describes a feature value of a point.The feature vector which describes a point can be conveniently modelled

Definition 4. 1 .
(i) Let (T, δ) be an IFTS.A singleton {x} ⊆ T is said to be IF closed if there exist an IF closed set C with M c (x) > 1 2 and N c (z) = 1, for all z ∈ T where z 6 = x.(ii) Let (T, δ) be an IFTS.A singleton {x} ⊆ T is said to be IF closed if there exist an IF open set O with M O (x) < 1 2 and M O (z) = 1, for all z ∈ T where z 6 = x.
). Definition 2.10.[5]Let(T, δ) and (Y, τ ) be two IFTSs and let f : T → Y be a map.Then f is said to be IF open if and only if the image of each IFS in δ is an IFS in τ .Definition 2.11.[3]Let {(T i , δ i )} i∈I be any indexed family of IFTSs.Then the product intuitionistic fuzzy topology (PIFT) Πδ i on ΠT i is the IFT generated by σ= {θ −1 i (U i ) | U i ∈ δ i , i ∈ I} as subbasis.The pair (ΠT i , Πδi) is called the product intuitionistic fuzzy topological space (PIFTS)., t 2 ∈ T and t 1 6 = t 2 , there exist