Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 604396, 7 pages

doi:10.1155/2012/604396

Research Article

Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Amit Kumar Singh¹ and Rekha Srivastava²

¹Electronics and Communication Sciences Unit, Indian Statistical Institute, Kolkata 700108, India

²Department of Applied Mathematics, Indian Institute of Technology (BHU), Varanasi 221005, India

Received 30 April 2012; Accepted 8 November 2012

Academic Editor: Mehmet Bodur

Copyright © 2012 Amit Kumar Singh and Rekha Srivastava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper we have studied separation axioms , in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functors and and observe that is left adjoint to .

1. Introduction

Fuzzy sets were introduced by Zadeh [1] in 1965 as follows: a fuzzy set in a nonempty set is a mapping from to the unit interval , and is interpreted as the degree of membership of in . Atanassov [2] generalized this concept and introduced intuitionistic fuzzy sets which take into account both the degrees of membership and of nonmembership subject to the condition that their sum does not exceed 1. Çoker [3] subsequently initiated a study of intuitionistic fuzzy topological spaces.

In this paper we have searched for appropriate definitions of the separation axioms , in intuitionistic fuzzy topological spaces.

Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Çoker [3], Bayhan and Çoker [4], and Lupianez [5]. In [4], the authors have given six possible definitions of Hausdorffness including that given in [3], and a comparative study has been done. In this paper we have introduced another definition which generalizes the corresponding definition in a fuzzy topological space given in [6]. Our definition is more general than those given in [3, 5], and it turns out to be equivalent to in [4].

-ness in an intuitionistic fuzzy topological space has been defined earlier in [4] in six possible ways. Out of those, we have chosen as it generalizes the most appropriate definition of -ness in a fuzzy topological space (cf. definition 5.1, [7]). We have also introduced a suitable definition of -ness in an intuitionistic topological space.

The appropriateness of the definitions has been established by proving several basic desirable results; for example, they satisfy hereditary, productive, and projective properties. We have also shown that the functor IF-Top BF-Top preserves these separation properties.

2. Preliminaries

Throughout denotes a nonempty set, denotes the unit interval , and and denote the intervals and , respectively. A fuzzy set in is a function from to . The collection of all fuzzy sets in is denoted by . For any , denotes the fuzzy complement of , and the constant fuzzy set in , taking value , is denoted by . A crisp subset of will be identified with its characteristic function. If , then will be identified with the fuzzy set in which takes the same value as if and zero if .

Definition 1 (Atanassov [2]). Let be a nonempty set. An intuitionistic fuzzy set (IFS, in short)   is an ordered pair of fuzzy sets in . Here = and , , respectively, denote the degree of membership and the degree of nonmembership of to the set and for each .

We identify an ordinary fuzzy set with the intuitionistic fuzzy set .

Definition 2 (Atanassov [2]). Let be a nonempty set and be given by and , respectively,

(a), (b), (c), (d), (e).

Definition 3 (Çoker [3]). Let be an arbitrary family of IFSs in . Then

(a), (b), (c), =.

Definition 4 (Çoker [3]). Let and be two nonempty sets and be a function. If and be IFSs in and , respectively, then

(a), (b).

It is easy to verify that and .

Definition 5 (Wong [8]). A fuzzy point in is a fuzzy set in taking value at and zero elsewhere, and and are, respectively, called the support and value of .

A fuzzy point is said to belong to a fuzzy set    if (cf. [6]).

Two fuzzy points are said to be distinct if their supports are distinct.

Definition 6. Let be a nonempty set and a fixed element in . If and are two fixed real numbers such that , then the IFS is called an intuitionistic fuzzy point IFP, in short in , and is called its support. Two IFPs are said to be distinct if their supports are distinct.

Let be an IFP in and be an IFS in . Then is said to belong to   , in short if , (cf. [9]).

We identify a fuzzy point in by the intuitionistic fuzzy point in .

Proposition 7. An intuitionistic fuzzy set in is the union of all intuitionistic fuzzy points belonging to .

The proof is on similar lines as in [10, Theorem  2.4] and hence is omitted.

Replacing fuzzy sets by intuitionistic fuzzy sets in Chang’s definition of a fuzzy topological space, we get the following.

Definition 8 (Çoker [3]). An intuitionistic fuzzy topology (IFT, in short) on a nonempty set is a family of IFSs in satisfying the following axioms:

(1), (2), (3) for any arbitrary family .

The pair   is called an intuitionistic fuzzy topological space (IFTS, in short), members of are called intuitionistic fuzzy open sets (IFOS, in short) in , and their complements are called intuitionistic fuzzy closed sets (IFCS, in short).

Definition 9. Let be an IFTS. A subfamily is called a base for if every can be written as a union of members of .

Proposition 10. Let be an IFTS, and then a subfamily is a base for if and only if for all and intuitionistic fuzzy point , such that .

The proof is easy omitted.

Definition 11. Let be an IFTS. Then a subfamily is called a subbase for if the family of finite intersections of members of forms a base for .

Given any collection of IFSs in , containing and , the set consisting of arbitrary unions of finite intersections of members of forms an IFT on . This is the smallest IFT on containing and is called the IFT generated by .

Definition 12 (S. J. Lee and E. P. Lee [10]). An IFS in an IFTS is called an intuitionistic fuzzy neighborhood (IFN, in short) of an IFP if such that .

Proposition 13. Let be an IFTS. Then an IFS in is an IFOS if and only if is an IFN of each of IFP .

The proof is on similar lines as in [10], Theorem  2.6 and hence is omitted.

Definition 14 (S. J. Lee and E. P. Lee [10]). A map between IFTSs is called intuitionistic fuzzy continuous if .

Definition 15 (Abu Safia et al. [12]). Let be a nonempty set and , be two fuzzy topologies on . Then is called a bifuzzy topological space (BFTS, in short).

A map between two BFTSs is said to be FP continuous if .

Definition 16 (Bayhan and Çoker [4]). Let and be IFSs in and , respectively, and then is the IFS in defined as follows

where , and , .

This definition can be extended to an arbitrary family of IFSs as follows.

If , is a family of IFSs in , then their product is defined as the IFS in given by

where inf , and ,.

Definition 17 (Bayhan and Çoker [4]). Let be two IFTSs, and then the product IFT on is defined as the IFT generated by where ,   are the projection maps, and the IFTS is called the product IFTS.

This definition can be extended to an arbitrary family of IFTSs as follows.

Let be a family of IFTSs. Then the product intuitionistic fuzzy topology on is the one having as a subbase where is the projection map. is called the product IFTS of the family .

Definition 18. A fuzzy topological space is called

(a) if , , such that either , , or , , (b) if , , such that , , , and , (c)   (Hausdorff) if for all pair of distinct fuzzy points in , such that , , and , (d)   (-Hausdorff) if for any pair of distinct fuzzy points , and    such that , and .

Here definitions (d), (c), (b), and (a) are from [5–7, 13], respectively.

Definition 19. Let be a BFTS. Then it is called

(a) if , such that , or , ,(b) if , and such that , and , , (c) if for all pair of distinct fuzzy points in , , such that , and .

Here definitions (a) and (b) are from [14], and (c) is from [15].

For the categorical concepts used here, we refer the reader to [16].

3. Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Definition 20. An IFTS is called

(a) if , , , such that , or , ,(b)(Bayhan and Çoker [4]). if , , , such that , , and ,(c)   (Hausdorff) if for all pair of distinct intuitionistic fuzzy points in , such that , and , (d) if for every pair of distinct intuitionistic fuzzy points in , and such that , and .

Example 21. Let and let , where and , then is an IFTS, and it is , , (Hausdorff) and .

We have and , but none of the implication are reversible.

Now we associate a BFTS with an IFTS and vice versa on parallel lines as in Bayhan and Çoker [11].

Let be an IFTS and such that , such that . It is easy to see that and are fuzzy topological spaces in Chang’s sense.

is called the bifuzzy topological space associated with the IFTS .

Proposition 22. Let be a BFTS and and . Then is an IFTS and , .

Proof. Clearly members of are intuitionistic fuzzy sets, and and belong to it. Now let , then = = . Further let , where is arbitrary}. Then . Thus is an IFTS.

Now let then . Therefore . Conversely let then such that , so . Thus . Similarly we can show that .

The IFTS is called the IFTS associated with the BFTS .

Proposition 23. Let and be two IFTSs and be IF-continuous. Then is FP-continuous (Here and are BFTSs associated with and , resp.).

Proof. Let be IF-continuous. To show that is FP-continuous, take then such that . Hence since is IF-continuous, , that is, . Further take then such that . Thus is FP-continuous.

Proposition 24. Let and be two BFTSs and and be the associated IFTSs respectively. Then is FP-continuous if and only if is IF-continuous.

Proof. Let be FP-continuous. To show that is IF continuous, take any , where , and . Now using FP-continuity of , and . So , that is, , showing that is IF-continuous.

The converse follows from the previous Proposition 23 in view of the fact that , and and .

The category of all BFTS together with FP-continuous functions will be denoted by BF-Top and the category of all IFTS together with IF-continuous function will be denoted by IF-Top.

Now we define IF-Top BF-Top as follows:

, morphism   and BF-Top IF-Top as follows:

, morphism .

It can be checked easily that and are covariant functors, and in view of Proposition 22 we have the following remark.

Remark 25. , the identity functor.

Theorem 26. The functor BF-Top IF-Top is left adjoint to the functor IF-Top BF-Top.

The proof is on parallel lines as in [11], Theorem  3.10 and hence is omitted.

Proposition 27. The following statements are equivalent in an IFTS :

(1) is ,(2) is intuitionistic fuzzy closed in , for all .

Proof. ⇒ We show that is intuitionistic fuzzy open in . Choose any IFP in then, . Hence IFOSs such that , and , . Now . Therefore in view of Proposition 13, is an IFOS.

⇒ Choose such that . Then and are IFCSs and hence , are IFOSs such that , , and showing that is .

Proposition 28. If an IFTS is , then its associated BFTS is .

Proof. Let be . Then , and such that , , , and . Now , .

Similarly, , , and . Now , and further , , , and showing that is a space.

Similarly it can be shown that if an IFTS is , then its associated BFTS is .

Theorem 29. Let be a family of IFTSs and be their product IFTS. Then is if and only if is , for all .

Proof. Let be   . To show that is , choose , . Let , then such that . Now since is , and such that , , , , and , , , . Now consider the basic IFOSs and where for and and when . Then , ), , ). Similarly we can show that and . Therefore is .

Conversly let be . To show that is , choose such that . Now consider , where , and the coordinate of are and , respectively. Then , therefore since is , and such that , , , and . Now consider the intuitionistic fuzzy points and . Then basic IFOSs and in such that and :

Similarly we can show that

Now is such that , . Further, since for , from (3) and (4), it follows that

Therefore

Thus and .  That is, . Similarly it can be shown that . Hence is .

The following theorem can be proved in a similar way.

Theorem 30. Let be a family of IFTSs and be their product IFTS. Then is if and only if is , for all .

Proposition 31. An IFTS is Hausdorff, and then its associated BFTS is Hausdorff.

Proof. Let be Hausdorff. To show that is Hausdorff, choose any two distinct fuzzy points , in . Now choose , and then and are distinct intuitionistic fuzzy points in . Since is Hausdorff, such that , and . Let , , then

and also we have and .

Now

From (8) we have and . Now we show that as follows.

We have,   in view of (9), .

Further,   (in view of (9), since .

Now take such that . If , then obviously , and if , then   (since .

Thus we have , such that , and , and hence is Hausdorff.

Definition 32. Let be an IFTS and . Then is called a subspace of where .

Proposition 33. If an IFTS is , , then its subspace,   is also , .

The proofs are easy and hence are omitted.

Proposition 34. The product IFTS of is initial with respect to the family of projections , that is, for any IFTS , a map is IF continuous if and only if the map is IF continuous for all  .

Proof. Since projection maps are IF continuous and composition of IF-continuous maps are IF-continuous, so is IF continuous .

Conversely, if is IF-continuous , then is IFO in    showing that inverse image of every subbasic IFOS in is IFO in which implies that is IF continuous.

Proposition 35. Let be a family of IFTSs, be the BFTS associated with , and be the BFTS associated with the product IFTS . Then and .

Proof. The product space is generated by where are projection maps. Let , and then .

Now members of are of the form:

So, members of are of the form , hence , and members of are of the form , hence .

Theorem 36. An IFTS is Hausdorff if and only if is IFCS in .

Proof. We show that is IFOS in . Choose any then and , . Now and are distinct intuitionistic fuzzy points in . Since is Hausdorff, IFOSs and in such that , and , that is, .

Now consider =, and then as shown below:

as = inf   (since both) and = sup    (since both).

Further, since and as and as .

Conversely, let be IFCS in . Then is IFOS in .  To show that is Hausdorff choose any two distinct intuitionistic fuzzy points , in then . Let max   =   and min   =  . Consider the intuitionistic fuzzy point in . Then and hence a basic IFOS in such that .

Now,   = inf , and   = sup implies that , similarly which implies that . Now we show that = . Since ,   =   inf = , and   = sup   =, . Thus = =.

Definition 37. A BFTS is called if for any two distinct fuzzy points and there exists , such that , and .

Proposition 38. An IFTS is , and then its associated BFTS is .

Proof. Let and be any two distinct fuzzy points in . Since is and , are distinct intuitionistic fuzzy points in , there exist and such that , and , . and and . Thus for fuzzy points and in , and . Further, since , we have   since . Hence is .

Theorem 39. Let be a family of IFTs and be their product IFTS. Then is Hausdorff if and only if each coordinate space is Hausdorff.

Proof. Let and be two distinct intuitionistic fuzzy points in . Let , then and hence such that . Consider the distinct intuitionistic fuzzy points and in . Since is Hausdorff, disjoint in such that , and . Let and , and then , and , . Equivalently, we have, , , that is, and , , that is, and , since .

Conversely, let be Hausdorff. To show that is Hausdorff, and consider two distinct intuitionistic fuzzy points and in , then . Now consider and in where for and the coordinate of are and , respectively. Since is Hausdorff, such that , and . Now consider the basic IFOSs and in such that and . Here the IFS except for finitely many ’s say and except for finitely many ’s say .

Now, inf ,

Similarly,

We claim that and also . If it is not so, then and , and and hence from (11) and (12) we have

implying that , , therefore which is a contradiction to the fact that .

So, , , and in view of (11) and (12) showing that , .

Now it remains to show that , that is, and . Let and in such that and . Consider in where for and .

Now

In view of (11), (12), and (14) we have implying that which is a contradiction. Hence . Now let where for and . Since ,

We have

Therefore from (11), (12), and (15) we have implying that , again a contradiction. Hence is Hausdorff.

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