The Scientific World Journal

Volume 2014 (2014), Article ID 674171, 9 pages

http://dx.doi.org/10.1155/2014/674171

Research Article

A Study of Frontier and Semifrontier in Intuitionistic Fuzzy Topological Spaces

Athar Kharal

National University of Sciences and Technology (NUST), H-12, Islamabad 44000, Pakistan

Received 29 August 2013; Accepted 30 October 2013; Published 11 February 2014

Academic Editors: F. J. Garcia-Pacheco and E. Momoniat

Copyright © 2014 Athar Kharal. 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.

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.

1. 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 of each element to a set , the degree of nonmembership 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 [12–14]. 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 Çoker [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.

2. Preliminaries

Definition 1 (see [16]). Let be a nonempty fixed set. An intuitionistic fuzzy set (briefly IFS) is an object of the form , where and are degrees of membership and nonmembership of each , respectively, and for each . A class of all the IFS’s in is denoted as IFS(). When there is no danger of confusion, an IFS may be written as .

Definition 2 (see [16]). Let be a nonempty set and , IFSs in . Then

(1) if and , for all ,(2) if and ,(3),(4) [15],(5) [15].

Definition 3 (see [15]). IFS’s and are defined as and , respectively.

Definition 4 (see [17]). Let and . An intuitionistic fuzzy point (IFP for short) of is an IFS of defined by

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 , 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 Çoker in [15], as follows.

Definition 6 (see [15]). An intuitionistic fuzzy topology (IFT for short) on a nonempty set is a family of IFSs in satisfying the following axioms:

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

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()).

Proposition 7 (see [19]). Let be an IFTS. Then the following hold:

(1),(2)If , then ,(3)If , then .

Definition 8 (see [15]). Let be an IFTS and an IFS in . Then the fuzzy interior and fuzzy closure of are denoted and defined as

Proposition 9 (see [15]). Let be an IFTS and , be IFSs in . Then the following properties hold:

(1), ,(2), ,(3), ,(4), ,(5), ,(6) ,(7), .

3. Intuitionistic Fuzzy Frontier

Definition 10 (see [19]). Let be an IFTS and let . Then is called an intuitionistic fuzzy frontier point (in short, IFFP) of if . The union of all the IFFPs of is called an IF frontier of and denoted by . It is clear that .

Proposition 11 (see [19]). For each , . However, the inclusion cannot be replaced by an equality.

Theorem 12. For an IFS in an IFTS , the following hold:

(1),(2)If is IFC then ,(3)If is IFO then ,(4).

Proof. (1)  .

(2) ; hence , if is IFC set in .

(3) is IFO implies is IFC. By , and by we get .

(4) .

Converse of and of Theorem 12 is, in general, not true as is shown by the following.

Example 13. Let be the IFTS defined by Çoker (Example 3.3 [15]). We choose IFSs and as

Then calculations give

but . Also

but .

Theorem 14. Let be an IFS in an IFTS . Then

(1),(2),(3),(4).

Proof. (1)  Since , therefore we have . This proves .

(2)  .

(3) .

(4) Consider

Example 15. To show that , , and of Theorem 14 are, in general, irreversible, we choose in Example 3.3 of [15],

Then calculations give

Remark 16. In general topology, the following hold:

Whereas in IF topology, we give counter-examples to show that these may not hold in general.

Example 17. In Example 3.3 of [15], we choose

then we have

We now investigate the expression . We first show that the equality does not hold and is in fact an irreversible inclusion.

Theorem 18. Let and be IFSs in an IFTS . Then .

Proof. Consider

The converse of Theorem 18 is in general not true, as is shown by the following.

Example 19. In Example 3.3 of [15], if we choose

then calculations give

Again if we choose

then

and choosing

we get

However, we have the following.

Theorem 20. For any IFSs and in an IFTS ,

Proof. Consider

Example 21. To show that the converse of Theorem 20 is in general not true, in Example 3.3 of [15] we choose

then calculations give

Theorem 22. For any IFS in an IFTS ,

(1),(2).

Remark 23. We checked 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 holds and its proof is sought. However, the converse of is, in general, not true as is shown by the following.

Example 24. In Example 3.3 of [15], if we choose

then, we have

Theorem 25 (see [13]). Let and be product related IFTSs. Then, for an IFS of and an IFS of ,

(1),(2).

Theorem 26. Let , be a family of product related IFTSs. If , then

Proof. We use Theorem 14   and Theorem 25 to prove this.

It suffices to prove this for . Consider

Definition 27 (see [19]). Let be an IFTS, and let . Then is called an intuitionistic Q-neighborhood (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 .

Definition 28 (see [19]). Let be an IFTS and let . Then is called an intuitionistic fuzzy adherence point (in short, IFAP) of if for each , .

Definition 29 (see [19]). Let be an IFTS and . Then is called an intuitionistic fuzzy accumulation point of if it satisfies the following conditions:

(1) is an IFAP of ,(2)if , then for each , and are quasicoincident at some point such that .

The union of all the intuitionistic fuzzy accumulation points of is called the derived set of and is denoted by . It is clear that .

Proposition 30 (see [19]). For any IFS in an IFTS , .

Corollary 31 (see [19]). Let . Then iff .

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 .

Theorem 33. Let be a mapping. Then the following are equivalent:

(1) is IF continuous,(2), for any IFS in .

Proof. Suppose that is IF continuous. Let be an IFS in . Since is IF closed in , is IF closed in . gives . Therefore, . Consequently, .

Suppose , where is an IFS in . Let be any IF closed set in . We show that is IF closed in . By our hypothesis, or gives or implies is IF closed in . Thus, is IF continuous.

Theorem 34. Let be an IF continuous mapping. Then , for any IFS in .

Proof. Suppose that is IF continuous. Let be an IFS in . Then

Therefore .

Lemma 35. Let and . Then .

Proof. Since implies , we have .

Definition 36 (see [15]). Let and be two IFTSs and , a function. Then is said to be fuzzy open if the image of each IFS in is in .

Theorem 37. Let be an IFO mapping and an IFS in . Then .

Proof. Suppose is IFO and is an IFS in . Put

Then is IF open and therefore is IF open in . This gives . From (32), we get . Then by Lemma 35, we have

Consequently, we have .

4. 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 -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:

(1),(2).

Proof. (1)  Let be an IFS in . From , it follows that is an IFSC set. Hence, . Since is IFSC, we have . Thus .

(2) This can be proved in a similar manner as .

Definition 41. Let be an IFS in IFTS . Then the intuitionistic fuzzy semifrontier of is defined as . Obviously, 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.

Theorem 43. For IFSs and in an IFTS , one has

(1),(2),(3),(4),(5),(6),(7),(8).

Proof. (5) and are both IFSO sets and , implies and . In Combination, or

(6) and imply . Conversely, and imply and is IFSO. But is the largest IFSO set contained in ; hence . This gives the equality.

(7) This follows easily from .

(8) Since ,

In the following theorem, –(5) are analogues of Theorem 12, and hence we omit their proofs.

Theorem 44. For an IFS in IFTS , the following hold:

(1),(2)if is IFSC, then ,(3)if is IFSO, then ,(4)let and (resp., ). Then (resp., ), where (resp., ) denotes the class of intuitionistic fuzzy semi-closed (resp. intuitionistic fuzzy semiopen) sets in ,(5),(6),(7).

Proof. (6) Since and , then we have

(7) .

The converse of , , , and of Theorem 44 is, in general, not true as is shown by the following.

Example 45. In Example 3.3 of [15], we choose

then calculations give

The following is an analogue of Theorem 14.

Theorem 46. Let be an IFS in IFTS . Then one has

(1),(2),(3),(4).

Example 47. To show that , , and of Theorem 46, are, in general, irreversible, in Example 3.3 of [15], we choose

then the calculations show

Remark 48. In general topology, the following hold:

Whereas, in Intuitionistic Fuzzy Topology, we give counter-examples to show that these may not hold in general.

Example 49. In Example 3.3 of [15], we choose

then calculations give

Theorem 50. Let and be IFSs in an IFTS . Then .

The converse of Theorem 50 is, in general, not true as is shown by the following.

Example 51. In Example 3.3 of [15], we choose

then calculations show

However, we have the following theorem which is an analogue of Theorem 20.

Theorem 52. For IFSs and in IFTS , one has

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.

Theorem 54. For an IFS in IFTS , one has

(1),(2).

Remark 55. As in the case of Theorem 22 we also do not know whether the equality in Theorem 54 holds or not. So we leave these as open problems. However, the converse of is, in general, not true as is shown by the following.

Example 56. In Example 3.3 of [15], we choose

then it is easy to see that .

Definition 57. An IFS in IFTS is called an intuitionistic fuzzy semi-Q-neighborhood of an IFP if there exists an IFSO set in , such that .

Theorem 58. An IFP if each intuitionistic fuzzy semi-Q-neighborhood of is quasicoincident with .

Proof. if for every fuzzy closed set , . This gives . Equivalently, if for every IFSO set , . That is, for every fuzzy open set satisfying , is not contained in , or . Thus, if every IFO Q-neighborhood of is quasi-coincident with .

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 . It is evident that .

Proposition 61. Let be an IFS in , then .

Proof. Let . Then from Theorem 58, . On the other hand, is either or ; for the latter case, by Definition 60, ; hence . The reverse inclusion is obvious.

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 is IFSO in for each IFO set in .

Theorem 64. Let be a function. Then the following are equivalent:

(1) is intuitionistic fuzzy semi-continuous,(2), for any IFS in .

Proof. Suppose that is intuitionistic fuzzy semi-continuous. Let be an IFS in . Since is IFC in , is IFSC in . gives . Therefore, . Consequently, .

Suppose . Letting be any IFC set in , we show that is IFSC in . By our hypothesis, or gives or implies is IFSC in . Thus, is intuitionistic fuzzy semi-continuous.

Theorem 65. Let be a intuitionistic fuzzy semi-continuous function. Then one has

for any IFS in .

Proof. Suppose that is intuitionistic fuzzy semi-continuous. Let be an IFS in .

Then

Therefore, .

Conflict of Interests

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

References

K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.

W.-L. Gau and D. J. Buehrer, “Vague sets,” IEEE Transactions on Systems, Man and Cybernetics, vol. 23, no. 2, pp. 610–614, 1993.

H. Bustince and P. Burillo, “Structures on intuitionistic fuzzy relations,” Fuzzy Sets and Systems, vol. 78, no. 3, pp. 293–303, 1996.

R. Biswas, “Intuitionistic fuzzy relations,” Bulletin for Studies and Exchanges on Fuzziness and Its Applications, vol. 70, pp. 22–29, 1997.

K. Atanassov and G. Gargov, “Intuitionistic fuzzy logic,” Comptes Rendus de l'Academie Bulgare des Sciences, vol. 43, no. 3, pp. 9–12, 1990.

K. Atanassov and C. Georgiev, “Intuitionistic fuzzy prolog,” Fuzzy Sets and Systems, vol. 53, no. 2, pp. 121–128, 1993.

S. K. De, R. Biswas, and A. R. Roy, “An application of intuitionistic fuzzy sets in medical diagnosis,” Fuzzy Sets and Systems, vol. 117, no. 2, pp. 209–213, 2001.

E. Szmidt and J. Kacprzyk, “Intuitionistic fuzzy sets in group decision making,” Notes on IFS, vol. 2, no. 1, pp. 11–14, 1996.

M.-H. Shu, C.-H. Cheng, and J.-R. Chang, “Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly,” Microelectronics Reliability, vol. 46, no. 12, pp. 2139–2148, 2006.

X. Tang, Spatial object modeling in fuzzy topological spaces with applications to land cover change in China [ITC Dissertation No.108], University of Twente, Enschede, The Netherlands, 2004.

H. Gurcay, D. Coker, and A. H. Es, “On fuzzy continuity in intuitionistic fuzzy topological spaces,” The Journal of Fuzzy Mathematics, vol. 5, pp. 365–378, 1997.

K. Hur and Y. B. Jun, “On intuitionistic fuzzy alpha-continuous mappings,” Honam Mathematical Journal, vol. 25, pp. 131–139, 2003.

J. K. Jeon, Y. B. Jun, and J. H. Park, “Intuitionistic fuzzy alpha-continuity and intuitionistic fuzzy precontinuity,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 19, pp. 3091–3101, 2005.

D. Çoker, “An introduction to intuitionistic fuzzy topological spaces,” Fuzzy Sets and Systems, vol. 88, no. 1, pp. 81–89, 1997.

K. Atanassov, “Review and new results on intuitionistic fuzzy sets,” Preprint IM- MFAIS-I-88, Sofia, 1988

S. J. Lee and E. P. Lee, “The category of intuitionistic fuzzy topological spaces,” Bulletin of the Korean Mathematical Society, vol. 37, no. 1, pp. 63–76, 2000.

C. L. Chang, “Fuzzy topological spaces,” Journal of Mathematical Analysis and Applications, vol. 24, no. 1, pp. 182–190, 1968.

K. Hur, J. H. Kim, and J. H. Ryou, “Intuitionistic fuzzy topological spaces,” Journal of the Korean Society of Mathematical Education B, vol. 11, no. 3, pp. 241–263, 2004.

N. Levine, “Semi-open sets and semi-continuity in topological spaces,” American Mathematical Monthly, vol. 70, pp. 36–41, 1963.

A. Rosenfeld, “Digital topology,” American Mathematical Monthly, vol. 86, pp. 621–630, 1979.

J. Cao, M. Ganster, and I. Reilly, “Submaximality, extremal disconnectedness and generalized closed sets,” Houston Journal of Mathematics, vol. 24, no. 4, pp. 681–688, 1998.

A. Rosenfeld, “Fuzzy digital topology,” Information and Control, vol. 40, no. 1, pp. 76–87, 1979.

K. K. Azad, “On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity,” Journal of Mathematical Analysis and Applications, vol. 82, no. 1, pp. 14–32, 1981.