A Study of Frontier and Semifrontier in Intuitionistic Fuzzy Topological Spaces

Notions of frontier and semifrontier in intuitionistic fuzzy topology have been studied and several of their properties, characterizations, and examples established. Many counter-examples have been presented to point divergences between the IF topology and its classical form. The paper also presents an open problem and one of its weaker forms.


Introduction
The intuitionistic fuzzy sets (IFSs) were introduced by Atanassov [1] as a generalization of fuzzy sets of Zadeh [2], where besides the degree of membership ( ) ∈ [0, 1] of each element ∈ to a set , the degree of nonmembership ( ) ∈ [0, 1] was also considered. IFS is a sufficiently generalized notion to include both fuzzy sets and vague sets. Fuzzy sets are IFSs but the converse is not necessarily true [1], whereas the notion of vague set defined by Gau and Buehrer [3] was proven by Bustince and Burillo [4] to be the same as IFS. IFSs have been found to be very useful in diverse applied areas of science and technology. In fact, there are situations where IFS theory is more appropriate to deal with [5]. IFSs have been applied to logic programming [6,7], medical diagnosis [8], decision making problems [9], microelectronic fault analysis [10], and many other areas.
Tang [11] has used fuzzy topology for studying land cover changes in China. Considering the inherent nature of Geographic Information Science (GIS) phenomena, it seems more suitable to study the problem of land cover changes using intuitionistic fuzzy topology. Tang has made a heavy use of the notion of fuzzy boundary. Thus, for recasting the GIS problem in terms of Intuitionistic Fuzzy Topology makes the study of intuitionistic fuzzy frontier imperative.
In this work we study the notion of frontier in IF topology and establish several of its properties, thus providing sufficient material for researchers to utilize these concepts fruitfully. The study of weaker forms of different notions of intuitionistic Fuzzy Topology is currently underway [12][13][14]. Using the notion of intutionistic fuzzy semisets, we also define the notion of intuitionistic fuzzy semifrontier and characterize intuitionistic fuzzy semicontinuous functions in terms of intuitionistic fuzzy semifrontier. We extend this study further in the last section and give many properties, characterizations, and examples pertaining to the generalized notion. It is noteworthy that all the counter examples given herein are constructed upon the intuitionistic fuzzy topological space defined by Ç oker [15]. In a developing field like IFS, it is interesting how the new theory differs from the old one. We have furnished two divergences from classical topology in Examples 17 and 49. An open problem and its semiversion are reported in Remarks 23 and 55.
Definition 4 (see [17]). Let , ∈ [0, 1] and + ≤ 1. An intuitionistic fuzzy point (IFP for short) ( , ) of is an IFS of defined by (1) In this case, is called the support of ( , ) and and are called the value and the nonvalue of ( , ) , respectively. An IFP ( , ) is said to belong to an IFS = ⟨ , ⟩ in , denoted by ( , ) ∈ if ≤ ( ) and ≥ ( ). Clearly an intuitionistic fuzzy point can be represented by an ordered pair of fuzzy points as follows: A class of all IFP's in is denoted as IFP( ).
Definition 5 (see [15]). If = ⟨ , ( ), ( )⟩ is an IFS in , then the preimage of under , denoted by −1 ( ), is the IFS in defined by If = ⟨ , ( ), ( )⟩ is an IFS in , then the image of under , denoted by ( ), is the IFS in defined by where The concept of fuzzy topological space, first introduced by Chang in [18], was generalized to the case of intuitionistic fuzzy sets by Ç oker in [15], as follows.
In this case, the pair ( , ) is called an intuitionistic fuzzy topological space (briefly, IFTS) and members of are called intuitionistic fuzzy open (briefly, IFO) sets. The complement of an IFO set is called an intuitionistic fuzzy closed (IFC) set in . Collection of all IFO (resp., IFC) sets in IFTS is denoted as IFO( ) (resp., IFC( )).

Intuitionistic Fuzzy Frontier
Definition 10 (see [19]). Let be an IFTS and let ∈ IFS( ). Then ( , ) ∈ IFP( ) is called an intuitionistic fuzzy frontier point (in short, IFFP) of if ( , ) ∈ Cl ⋂ Cl . The union of all the IFFPs of is called an IF frontier of and denoted by Fr . It is clear that Fr = Cl ⋂ Cl .

Theorem 12.
For an IFS in an IFTS , the following hold: (3) is IFO implies is IFC. By (2), Fr ⊆ and by (1) we get Fr ⊆ .
Converse of (2) and (3) of Theorem 12 is, in general, not true as is shown by the following.

Theorem 14. Let be an IFS in an IFTS . Then
Fr Int ⊆ Fr , Then calculations give Remark 16. In general topology, the following hold: We now investigate the expression Fr ( ⋃ ). We first show that the equality Fr ( ⋃ ) = Fr ⋃ Fr does not hold and is in fact an irreversible inclusion.

Theorem 18. Let and be IFSs in an IFTS . Then
The Scientific World Journal The converse of Theorem 18 is in general not true, as is shown by the following.
However, we have the following.

Theorem 20. For any IFSs and in an IFTS ,
Proof. Consider  Remark 23. We checked (2) of Theorem 22 on a large number of IFTSs, no counter-example could be found to establish the irreversibility of inequality. Therefore, it is conjectured that the equality in (2) holds and its proof is sought. However, the converse of (1) is, in general, not true as is shown by the following.
Theorem 25 (see [13]). Let and be product related IFTSs. Then, for an IFS of and an IFS of , Proof. We use Theorem 14 (1) and Theorem 25 to prove this. It suffices to prove this for = 2. Consider Definition 27 (see [19]). Let ( , ) be an IFTS, ∈ IFS( ) and let ( , ) ∈ IFP( ). Then is called an intuitionistic Qneighborhood (in short, IQN) of ( , ) if there is a ∈ such that ( , ) ⊂ . The family of all the IQNs of ( , ) is called the system of IQNs of ( , ) and denoted by N IQ ( ( , ) ).
The union of all the intuitionistic fuzzy accumulation points of is called the derived set of and is denoted by . It is clear that ⊂ Cl .
Definition 32 (see [15]). Let ( , ) and ( , ) be two IFTSs and : → , a function. Then is said to be intuitionistic fuzzy continuous if the preimage of each IFS in is in . (2) ⇒ (1) Suppose ( ) ⊆ Cl ( ), where is an IFS in . Let be any IF closed set in . We show that −1 ( ) is IF closed in . By our hypothesis, The Scientific World Journal Therefore Fr −1 ( ) ⊆ −1 (Fr ).

Intuitionistic Fuzzy Semifrontier
Levine [20] generalized the notion of open sets as semiopen sets. His impetus for the generalization was to develop a wider framework for the study of continuity and its different variants. Interestingly, his work also found application in the field of digital topology [21], though it was never in sight at the time of inception of semitype notions (technically known as weaker notions). For example, it was found that digital line is a 1/2 -space [22], which is a weaker separation axiom based upon semiopen sets. Fuzzy digital topology [23] was introduced by Rosenfeld, which demonstrated the need for the fuzzification of weaker forms of notions of classical topology. Azad [24] carried out this fuzzification in 1981, and thus initiated the study of the concepts of fuzzy semiopen and fuzzy semiclosed sets. Intutionistic Fuzzy Topology, being a relatively new field also followed the trajectory of its nearest analogue: fuzzy topology. Thus study of weaker forms of different notions in the settings of Intuitionistic Fuzzy Topology is currently a very active area of research [13,14]. In this section, we generalize the definitions and results of intuitionistic fuzzy frontier in the intuitionistic fuzzy semisettings.
Definition 38 (see [12]). An IFS in an IFTS ( , ) is called an intuitionistic fuzzy semiopen set (IFSOS) if An IFS is called an intuitionistic fuzzy semiclosed set if the complement of is an IFSOS.
Definition 39. The semiclosure and semi-interior of an IFS in an IFTS ( , ) are denoted and defined as Theorem 40. For an IFS in IFTS , the following hold: (2) This can be proved in a similar manner as (1).
Definition 41. Let be an IFS in IFTS . Then the intuitionistic fuzzy semifrontier of is defined as Fr = Cl ⋂ Cl . Obviously, Fr is an IFSC set.
Remark 42. In the following theorems, we note that almost all the properties related to intuitionistic fuzzy semi-interior, intuitionistic fuzzy semi-closure and intuitionistic fuzzy semifrontier are analogous to their counterparts in Intuitionistic Fuzzy Topology, and hence proofs of most of them are not given.
In the following theorem, (1)-(5) are analogues of Theorem 12, and hence we omit their proofs.
The following is an analogue of Theorem 14.

Theorem 46. Let be an IFS in IFTS . Then one has
(1) Fr = Cl − Int , (2) Fr Int ⊆ Fr , then the calculations show Remark 48. In general topology, the following hold: However, we have the following theorem which is an analogue of Theorem 20.

Corollary 53. For IFSs and in IFTS , one has
The analogue of Theorem 22 is the following theorem, the proof of which is easy to establish.
Remark 55. As in the case of Theorem 22(2) we also do not know whether the equality in Theorem 54(2) holds or not. So we leave these as open problems. However, the converse of (1) is, in general, not true as is shown by the following.  Definition 59. An IFP is called a semiadherence point of an IFS if every intuitionistic fuzzy semi-Q-neighborhood of is quasi-coincident with .
Definition 60. An IFP is called a semiaccumulation point of an IFS if is a semi-adherence point of and every semi-Q-neighborhood of and is quasi-coincident at some point different from supp( ), whenever ∈ . The union of all the semi-accumulation points of is called the intuitionistic fuzzy semiderived set of , denoted as sd . It is evident that sd ⊆ Cl .
Proposition 61. Let be an IFS in , then Cl = ⋃ .

Corollary 62. For any IFS in an IFTS , is IFSC if
⊆ .
Definition 63. Let : → be a function from an IFTS to another IFTS . Then is said to be an intuitionistic fuzzy semicontinuous function if −1 ( ) is IFSO in for each IFO set in .
Theorem 65. Let : → be a intuitionistic fuzzy semicontinuous function. Then one has for any IFS in .
Proof. Suppose that is intuitionistic fuzzy semi-continuous. Let be an IFS in .

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.