The Scientific World Journal
Volume 2015 (2015), Article ID 869740, 5 pages
http://dx.doi.org/10.1155/2015/869740

Research Article

On Intuitionistic Fuzzy -Almost Compactness and -Nearly Compactness

R. Renuka and V. Seenivasan

Department of Mathematics, University College of Engineering Panruti (A Constituent College of Anna University Chennai), Panruti, Tamil Nadu 607 106, India

Received 19 May 2015; Accepted 18 June 2015

Academic Editor: Abdelalim A. Elsadany

Copyright © 2015 R. Renuka and V. Seenivasan. 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.

The concept of intuitionistic fuzzy β-almost compactness and intuitionistic fuzzy β-nearly compactness in intuitionistic fuzzy topological spaces is introduced and studied. Besides giving characterizations of these spaces, we study some of their properties. Also, we investigate the behavior of intuitionistic fuzzy β-compactness, intuitionistic fuzzy β-almost compactness, and intuitionistic fuzzy β-nearly compactness under several types of intuitionistic fuzzy continuous mappings.

1. Introduction

As a generalization of fuzzy sets introduced by Zadeh [1], the concept of intuitionistic fuzzy set was introduced by Atanassov [2]. Coker [3] introduced intuitionistic fuzzy topological spaces. Later on, Coker [3] introduced fuzzy compactness in intuitionistic fuzzy topological spaces. In [4], the concept of intuitionistic fuzzy -compactness was defined. In this paper some properties of intuitionistic fuzzy -compactness were investigated. We use the finite intersection property to give characterization of the intuitionistic fuzzy -compact spaces. Also we introduce intuitionistic fuzzy -almost compactness and intuitionistic fuzzy -nearly compactness and established the relationships between these types of compactness.

2. Preliminaries

Definition 1 (see [2]). Let be a nonempty fixed set and the closed interval . An intuitionistic fuzzy set (IFS) A is an object of the following form:

where the mappings and denote the degree of membership, namely, , and the degree of nonmembership, namely, , for each element to the set , respectively, and for each .

Definition 2 (see [2]). Let and be IFSs of the forms and . Then

(i) if and only if and ;(ii);(iii);(iv).

We will use the notation instead of .

Definition 3 (see [2]). Consider and .

Let such that . An intuitionistic fuzzy point (IFP) is intuitionistic fuzzy set defined by

Definition 4 (see [3]). An intuitionistic fuzzy topology (IFT) in Coker’s sense on a nonempty set is a family of intuitionistic fuzzy sets in satisfying the following axioms:

(i);(ii), for any ;(iii) for any arbitrary family .

In this paper by and or simply by and , we will denote the intuitionistic fuzzy topological spaces (IFTSs). Each IFS which belongs to is called an intuitionistic fuzzy open set (IFOS) in . The complement of an IFOS A in is called an intuitionistic fuzzy closed set (IFCS) in .

Let and be two nonempty sets and let be a function.

If is an IFS in , then the preimage of under is denoted and defined by . Since are fuzzy sets, we explain that and .

Definition 5 (see [5]). Let be an IFP in IFTS . An IFS in is called an intuitionistic fuzzy neighborhood (IFN) of if there exists an IFOS in such that .

Definition 6 (see [3]). Let be an IFTS and let be an IFS in . Then the intuitionistic fuzzy closure and intuitionistic fuzzy interior of are defined by

(i);(ii).

It can be also shown that is an IFCS, is an IFOS in , and is an IFCS in if and only if ; is an IFOS in if and only if .

Proposition 7 (see [3]). Let be an IFTS and let and be IFSs in . Then the following properties hold:

(i), ;(ii).

Definition 8 (see [6]). An IFS in an IFTS is called an intuitionistic fuzzy -open set (IFOS) if . The complement of an IFOS A in IFTS is called an intuitionistic fuzzy -closed set (IFCS) in .

Definition 9 (see [6]). Let be a mapping from an IFTS into an IFTS . The mapping is called

(i)intuitionistic fuzzy continuous if and only if is an IFOS in , for each IFOS in ;(ii)intuitionistic fuzzy -continuous if and only if is an IFOS in , for each IFOS in .

Definition 10 (see [4]). Let be an IFTS and let be an IFS in . Then the intuitionistic fuzzy -closure and intuitionistic fuzzy -interior of are defined by

(i);(ii).

Definition 11 (see [7]). A fuzzy function is called fuzzy -irresolute if inverse image of each fuzzy -open set is fuzzy -open.

Definition 12 (see [4]). A function from an intuitionistic fuzzy topological space to another intuitionistic fuzzy topological space is said to be intuitionistic fuzzy -irresolute if is an IFOS in for each IFOS in .

Definition 13 (see [3, 4]). Let be an IFTS. A family of intuitionistic fuzzy open sets (intuitionistic fuzzy -open sets) in satisfies the condition which is called an intuitionistic fuzzy open (intuitionistic fuzzy -open) cover of . A finite subfamily of an intuitionistic fuzzy open (intuitionistic fuzzy -open) cover of which is also an intuitionistic fuzzy open (intuitionistic fuzzy -open) cover of is called a finite subcover of .

Definition 14 (see [3]). An IFTS is called intuitionistic fuzzy compact if each fuzzy open cover of has a finite subcover for .

Definition 15 (see [8]). An IFTS is called intuitionistic fuzzy almost compact (IF almost compact) if, for every IF open cover of , there exists a finite subfamily such that .

Definition 16 (see [4]). An IFTS is said to be intuitionistic fuzzy -compact (IF -compact) if every IF -open cover of has a finite subcover.

Definition 17 (see [6]). Let and be two IFTSs. A function is said to be intuitionistic fuzzy weakly continuous (IF weakly continuous) if, for each IFOS in , .

3. Intuitionistic Fuzzy -Compactness

Definition 18. An IFTS of is said to be IF -compact relative to if, for every collection of IF -open subsets of such that , there exists a finite subset of such that .

Definition 19. An IFTS of is said to be IF -compact if it is IF -compact as a subspace of .

Definition 20. A family of IF -closed sets has the finite intersection property (in short FIP) if, for any subset of , .

Theorem 21. For an IFTS the following statements are equivalent.

(i) is IF -compact.(ii)Any family of IF -closed subsets of satisfying the FIP has a nonempty intersection.

Proof. Let be IF -compact space and let be a family of IF -closed subsets of satisfying the FIP. Suppose . Then . Since is a collection of IF -open sets cover , then from IF -compactness of there exists a finite subset of such that . Then which gives a contradiction and therefore . Thus .
Let be a family of IF -open sets cover . Suppose that for any finite subset of we have . Then . Hence satisfies the FIP. Then, by hypothesis, we have which implies that and contradicts that is an IF -open cover of . Hence our assumption is wrong. Thus which implies that is IF -compact. Thus .

Theorem 22. An intuitionistic fuzzy -closed subset of an intuitionistic fuzzy -compact space is intuitionistic fuzzy -compact relative to .

Proof. Let be an IF -closed subset of . Let be cover of by IF -open sets in . Then the family is an IF -open cover of . Since is IF -compact, there is a finite subfamily of IF -open cover, which also covers . If this cover contains , we discard it. Otherwise leave the subcover as it is. Thus we obtained a finite IF -open subcover of . So is IF -compact relative to .

Theorem 23. Let and be intuitionistic fuzzy topological spaces and let be intuitionistic fuzzy -irresolute, surjective mapping. If is IF -compact space then so is .

Proof. Let be intuitionistic fuzzy -irresolute mapping of an intuitionistic fuzzy -compact space onto an IFTS . Let be any intuitionistic fuzzy -open cover of . Then is collection of intuitionistic fuzzy -open sets which covers . Since is intuitionistic fuzzy -compact, there exists a finite subset of such that subfamily of covers . It follows that is a finite subfamily of which covers . Hence is intuitionistic fuzzy -compact.

Theorem 24. Let and be intuitionistic fuzzy topological spaces and let be intuitionistic fuzzy -irresolute mapping. If is IF -compact relative to then is IF -compact relative to .

Proof. Let be a family of IF -open cover of such that . Then . Since is IF -irresolute, is IF -open cover of . And is IF -compact in ; there exists a finite subset of such that . Hence . Thus is IF -compact relative to .

Theorem 25. An IF -continuous image of IF -compact space is IF compact.

Proof. Let be an IF -continuous from an IF -compact space onto IFTS . Let be IF open cover of . Then is IF -open cover of . Since is IF -compact, there exists finite subset of such that finite family covers . Since is onto, is a finite cover of . Hence is IF compact.

Definition 26. Let and be two intuitionistic fuzzy topological spaces. A mapping is said to be intuitionistic fuzzy strongly -open if is IFOS of for every IFOS of .

Theorem 27. Let be an IF strongly -open, bijective function and is IF -compact; then is IF -compact.

Proof. Let be IF -open cover of , and then is IF -open cover of . Since is IF -compact, there is a finite subset of such that finite family covers . But and therefore is IF -compact.

4. Intuitionistic Fuzzy -Almost Compactness and Intuitionistic Fuzzy -Nearly Compactness

In this section we investigate the relationships between IF -compactness, IF -almost compactness, and IF -nearly compactness.

Definition 28. An IFTS is said to be IF -almost compactness if and only if, for every family of IF -open cover of , there exists a finite subset of such that .

Definition 29. An IFTS is said to be IF -nearly compactness if and only if, for every family of IF -open cover of , there exists a finite subset of such that .

Definition 30. An IFTS is said to be IF -regular if, for each IF -open set , .

Theorem 31. Let be IFTS. Then IF -compactness implies IF -nearly compactness which implies IF -almost compactness.

Proof. Let be IF -compact. Then for every IF -open cover of , there exists a finite subset . Since is an IFOS, for each , for each . for each . Here . Thus which implies that is IF -nearly compactness. Now let be IF -nearly compact. Then for every IF -open cover of , there exists a finite subset . Since for each , . Thus . Hence is IF -almost compact.

Remark 32. Converse implications in theorem are not true in general.

Example 33. Let be a nonempty set. Then is IFTS, where , , where is defined by , . The collection is IF -open cover of . But no finite subset of covers . Hence is not IF -compact. But for . Thus there exists a finite subfamily for such that . Thus is IF -almost compactness. Also for . Thus there exists a finite subfamily for such that . Thus is IF -nearly compactness.

Theorem 34. Let be IFTS. If is IF -almost compact and IF -regular then is IF -compact.

Proof. Let be IF -open cover of such that . Since is IF -regular, for each . Since and is IF -almost compact there exists a finite set of such that . But . We have . Thus, . Hence is IF -compact.

Theorem 35. Let be IFTS. If is IF -nearly compact and IF -regular then is IF -compact.

Proof. Let be IF -open cover of such that . Since is IF -regular, for each . Since and is IF -nearly compact there exists a finite set of such that . But . We have . Thus, . Hence is IF -compact.

Theorem 36. An IFTS is IF -almost compact, if and only if, for every family of IF -open sets having the FIP, .

Proof. Let be a family of IF -open sets having the FIP. Suppose that and then . Since is IF -almost compact, there exists a finite subset of such that . This implies that . Thus, . But . This implies that which is in contradiction with FIP of the family. Conversely, let be a family of IF -open sets such that . Suppose that there exists no finite subset of such that . Since has the FIP then . This implies . Hence . Since for each , which is in contradiction with .

Theorem 37. Let and be IFTS and let be intuitionistic fuzzy -irresolute, surjective mapping. If is IF -almost compact space then so is .

Proof. Let be intuitionistic fuzzy -irresolute mapping of an intuitionistic fuzzy -compact space onto an IFTS . Let be any intuitionistic fuzzy -open cover of . Then is an intuitionistic fuzzy -open cover of . Since is intuitionistic fuzzy -almost compact, there exists a finite subset of such that . And . But and from the surjectivity of , . So . Thus . Hence is IF -almost compact.

Theorem 38. Let and be intuitionistic fuzzy topological spaces and let be intuitionistic fuzzy -continuous, surjective mapping. If is IF -almost compact space then is IF almost compact.

Proof. Let be any intuitionistic fuzzy open cover of . Then is an intuitionistic fuzzy -open cover of . Since is intuitionistic fuzzy -almost compact, there exists a finite subset of such that . And from the surjectivity of , which implies that . Hence is IF almost compact.

Definition 39. Let and be two intuitionistic fuzzy topological spaces. A function is said to be intuitionistic fuzzy -weakly continuous (IF -weakly continuous) if, for each IFOS in , .

Theorem 40. A mapping from an IFTS to an IFTS is IF strongly -open if and only if .

Proof. If is IF strongly -open mapping then is an IFOS in for IFOS in . Hence . Thus .
Conversely, let be IFOS in and then . Then by hypothesis, . This implies that is IFOS in .

Theorem 41. Let and be IFTS and let be intuitionistic fuzzy -weakly continuous, surjective mapping. If is IF -compact space then is IF -almost compact.

Proof. Let be IF -open cover of such that . Then . is IF -compact, and there exists a finite subset of such that . Since is IF -weakly continuous, . This implies that . Thus . Since is surjective, . Hence . Hence is IF -almost compact.

Theorem 42. Let and be intuitionistic fuzzy topological spaces and let be intuitionistic fuzzy -irresolute, surjective, and strongly -open mapping. If is IF -nearly compact space then so is .

Proof. Let be any intuitionistic fuzzy -open cover of . Since is IF -irresolute, then is an intuitionistic fuzzy -open cover of . Since is IF -nearly compact, there exists a finite subset of such that . Since is surjective, . Since is IF strongly -open, for each . Since is IF -irresolute, then . Hence we have             . Thus . Hence is IF -nearly compact.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to thank the referee for his suggestions and corrections which helped to improve this paper.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.

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

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

R. Renuka and V. Seenivasan, “On intuitionistic fuzzy pre--irresolute functions,” Scientia Magna, vol. 9, no. 2, pp. 93–102, 2013.

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.

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

E. Ekici and J. H. Park, “On fuzzy irresolute functions,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 5, no. 2, pp. 164–168, 2005.

D. Coker and A. H. Es, “On fuzzy compactness in intuitionistic fuzzy topological spaces,” Journal of Fuzzy Mathematics, vol. 3, no. 4, pp. 899–909, 1995.