International Journal of Mathematics and Mathematical Sciences

Volume 2011, Article ID 853870, 10 pages

http://dx.doi.org/10.1155/2011/853870

## -Closed and -Closed in -Topological Spaces

^{1}Department of Mathematics, Mansoura University, P.O. Box 35516, Mansoura 35516, Egypt^{2}Department of Mathematics, Jazan University, P.O. Box 100, Jaza, Damad, Saudi Arabia

Received 8 June 2010; Accepted 14 September 2010

Academic Editor: Ayman Badawi

Copyright © 2011 M. E. El-Shafei and A. H. Zakari. 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

We investigate various classes of generalized closed fuzzy sets in -topological spaces, namely, -closed fuzzy sets and -closed fuzzy sets. Also, we introduce a new separation axiom of the -topological spaces, and we prove that every -space is a -space. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy mappings.

#### 1. Introduction

In 1970, Levine [1] introduced the notion of generalized closed sets in topological spaces as a generalization of closed sets. Since then, many concepts related to generalized closed sets were defined and investigated. In 1997, Balasubramanian and Sundaram [2] introduced the concepts of generalized closed sets in fuzzy setting. Also, they studied various generalizations fuzzy continues mappings.

Recently, El-Shafei and Zakari [3–5] introduced new types of generalized closed fuzzy sets in -topological spaces and studied many of their properties. Also, they studied various generalizations fuzzy continues mappings.

In the present paper, we introduce the concepts of -closed fuzzy sets and -closed fuzzy sets and study some of their properties. Also, we introduce the concept of -space. Moreover, we introduce and study the concepts of two new classes of fuzzy mappings, namely, fuzzy -continuous mappings and fuzzy -irresolute mappings.

#### 2. Preliminaries

Let be a set and the unit interval. A fuzzy set in is an element of the set of all functions from into . The family of all fuzzy sets in is denoted by . A fuzzy singleton is a fuzzy set in defined by , for all , . The set of all fuzzy singletons in is denoted by . For every and , we define if and only if . A fuzzy set is called quasicoincident with a fuzzy set , denoted by , if and only if there exists such that . If is not quasicoincident with , then we write . By , , , , and , we mean the fuzzy closure of , the fuzzy interior of , the complement of , the class of all open neighborhoods of , and the class of all open -neighborhoods of , respectively.

*Definition 2.1 (see [6, 7]). *A fuzzy subset of a -topological space is called(i)regular open if and only if ,(ii)preopen if and only if .

The complement of a regular open (resp. preopen) fuzzy set is called a regular closed (resp. preclosed).

*Definition 2.2 (see [8, 9]). *Let be a -topological space, , and . Then,(i)the -closure of , denoted by , is defined by if and only if for each ,(ii)the -closure of denoted by , is defined by if and only if for each ,(iii) is called -closed (resp. -closed) if and only if (resp. ).

*Definition 2.3 (see [9]). *Let be a -topological space and . Then,(i)the family is called an open -cover of if and only if for every , there exists such that ,(ii) is called a -set if and only if every open -cover of has a finite subcover.

*Definition 2.4 (see [2–4]). *Let be a -topological space. A fuzzy set is called(i)a generalized closed (-closed, for short) if and only if whenever and is open fuzzy set,(ii)a -generalized closed (-closed, for short) if and only if whenever and is open fuzzy set,(iii)a -generalized closed (-closed, for short) if and only if whenever and is open fuzzy set.

*Definition 2.5 (see [2, 4, 6, 10]). *A -topological space is called(i) if and only if implies that there exist and such that ,(ii) or -regular if and only if , is closed fuzzy set implies that there exist and , such that ,(iii) if and only if every -closed fuzzy set in is closed,(iv) if and only if every -closed fuzzy set in is -closed,(v)fuzzy weakly Hausdorff (, for short) if implies that there exists regular open fuzzy set such that ,(vi)fuzzy semiregular if and only if the collection of all regular open fuzzy sets in forms a base for the -topology ,(vii)a fuzzy partition space if and only if every open fuzzy subset is closed.

*Definition 2.6 (see [2–4, 11]). *A fuzzy mapping is called(i)fuzzy generalized continuous (fuzzy -continuous, for short) if and only if is -closed in for any closed fuzzy set in ,(ii) fuzzy -generalized continuous (fuzzy -continuous, for short) if and only if is -closed in for any closed fuzzy set in ,(iii) fuzzy -generalized continuous (fuzzy -continuous, for short) if and only if is -closed in for any closed fuzzy set in ,(iv) fuzzy -continuous if the inverse image of every -open fuzzy set in is -open in ,(v) fuzzy -open (fuzzy -open, for short) if and only if is -open in for any -open fuzzy set in ,(vi)fuzzy -closed (fuzzy -closed, for short) if and only if is -closed in for any -closed fuzzy set in .

Theorem 2.7 (see [3]). *A fuzzy subset of an -fts is -closed if and only if it is -closed.*

Theorem 2.8 (see [3]). *Let be a -topological space. Then, the following conditions are equivalent:*(i)* is -space,*(ii)*for each -set , ,*(iii)*for each , .*

Theorem 2.9 (see [3, 4]). *Let be a -topological space and be a preopen. Then, is -closed (resp.-closed) if and only if it is -closed.*

Theorem 2.10 (see [3]). *Let be a -topological space and . Then,*(i)*,*(ii)*.*

Theorem 2.11 (see [4]). *Let be a fuzzy semiregular space and . Then, *(i)* is -closed if and only if is -closed,*(ii)* If, in addition, is , then is -closed if and only if is closed.*

Theorem 2.12 (see [4]). *Let be an -space and be a -set. Then, is-closed if and only if it is -closed.*

Theorem 2.13 (see [4]). *Let be a fuzzy partition space and . Then, is -closed if and only if it is -closed.*

Theorem 2.14 (see [4]). *Let be a -topological space and . Then,*(i)*,*(ii)*.*

Theorem 2.15 (see [4]). *A -topological space is -space if for every either is -open or is closed.*

Theorem 2.16 (see [4]). *Let be a -topological space. Then, the following conditions are equivalent: *(i)* is an -space,*(ii)* for each .*

#### 3. -Closed Fuzzy Sets

In this section, we introduce the concept of weakly -generalized closed fuzzy sets, and we study some of their properties.

*Definition 3.1. *A fuzzy subset of a -topological space is said to be weakly -generalized closed (-closed, for short) if and only if whenever and is -open fuzzy set.

The complement of a -closed fuzzy set is called -open.

Theorem 3.2. *Let be a -topological space. Then,*(i)*Every -closed fuzzy set is -closed,*(ii)*Every -closed fuzzy set is -closed.*

*Proof. *Obvious.

From the above discussion, we introduce the following diagram.(3.1)

None of these implications is reversible as the following examples show.

*Example 3.3. *Let and . If , then is -closed fuzzy set but not -closed.

*Example 3.4. *Let and . If , then is -closed, since the only -open superset of is . But is not -closed, since and .

Theorem 3.5. *A fuzzy subset of a -topological space is -closed if for every such that , one has .*

*Proof. *Let be -open and . If , then by assumption, . Hence, there exists such that . Put . Then, and . Thus, for each . Since , then and so . Thus, is -closed.

Theorem 3.6. *Let be a -topological space and . Then, is -closed if there is not any -closed fuzzy set such that and .*

*Proof. *Suppose that is not -closed. Then, there exists -open fuzzy set such that and . Put . Then, there exists -closed fuzzy set such that and . This is a contradiction.

Theorem 3.7. *Let be an -space and be a -set and -closed. Then, is -closed.*

*Proof. *Suppose that is an -space and is a -set in . If is -closed, then by Theorem 2.8 is -closed and hence -closed.

Theorem 3.8. *Let be a -topological space and be a preopen and -closed. Then, is -closed. *

*Proof. *It is an immediate consequence of Theorems 2.9 and 3.2.

Theorem 3.9. *Let be an -space and be a -closed. Then, is -closed.*

*Proof. *It is an immediate consequence of Theorems 2.7 and 3.2.

Theorem 3.10. *A finite union of -closed fuzzy sets, is always -closed fuzzy set.*

*Proof. *Suppose that are -closed fuzzy sets and let such that . Since and are -closed, then we have and by Theorem 2.10(i) . Hence, is -closed.

#### 4. -Closed Fuzzy Sets

In this section, we introduce the concept of weakly -generalized closed fuzzy sets, and we study some of their properties. Also, we introduce the notion of -space, and we prove that every -space is a -space.

*Definition 4.1. *A fuzzy subset of -topological space is said to be weakly -generalized closed (-closed, for short) if and only if whenever and is -open fuzzy set.

The complement of a -closed fuzzy set is called -open.

Theorem 4.2. *Let be a -topological space. Then,*(i)*Every -closed fuzzy set is -closed,*(ii)*Every -closed fuzzy set is -closed.*

*Proof. *Obvious.

From the above discussion, we introduce the following diagram.(4.1)

None of these implications is reversible as the following examples show.

*Example 4.3. *Let and . If , then is -closed, since the only -open superset of is . But is not -closed, since and .

*Example 4.4. *Let and . A fuzzy subset is -closed and hence -closed, but it is not -closed.

Theorem 4.5. *A fuzzy subset of a -topological space is -closed if and only if for every such that one has .*

*Proof. *Let and suppose that . Since is -closed, then it is easy to observe that which implies that . This is a contradiction.

The converse is similar to the proof of Theorem 3.5.

Theorem 4.6. *Let be a -topological space and . Then, is -closed if and only if there is not any -closed fuzzy set such that and .*

*Proof. *Suppose that there is a -closed fuzzy set such that and . Then, there exists some such that . Since is -closed, then by using Theorem 4.5, and hence . Since is -closed, then we have . This is a contradiction.

The converse is similar to the proof of Theorem 3.6.

Theorem 4.7. *Let be a fuzzy semiregular space and . Then,*(i)* is -closed if and only if it is -closed,*(ii)*If, in addition, is -space, then is -closed if and only if it is closed.*

*Proof. *(i) Since is semiregular space, then , and so is -closed if and only if it is -closed.

(ii) From (i), Theorem 2.11, and by -ness, the result is given.

Theorem 4.8. *Let be an -space and be a -set and -closed. Then, is -closed.*

*Proof. *Suppose that is an -space and is a -set in . If is -closed, then by Theorem 2.12 is -closed and hence -closed.

Theorem 4.9. *Let be a -topological space and be a preopen and -closed. Then, is -closed.*

*Proof. *It is an immediate consequence of Theorems 2.9 and 4.2.

Theorem 4.10. *Every fuzzy subset of a fuzzy partition space is -closed.*

*Proof. *Let be a fuzzy partition space, and let be a fuzzy subset of . Then, by Theorem 2.13, is -closed and hence, by Theorem 4.2, is -closed.

Theorem 4.11. *A finite union of -closed fuzzy sets is always -closed fuzzy set.*

*Proof. *Similar to the proof of Theorem 3.10.

The following example shows that the finite intersection of -closed fuzzy set may fail to be -closed fuzzy set.

*Example 4.12. *Let . Define as follows:

Consider the -topology . It is clear that and are -closed fuzzy sets. But is not -closed.

*Definition 4.13. *A -topological space is called -space if and only if every -closed fuzzy set is -closed.

Theorem 4.14. *Every -space is -space.*

*Proof. *It is an immediate consequence of Theorem 4.2(ii).

Theorem 4.15. *A -topological space is -space if for every either is -open or -closed.*

*Proof. *Let be -closed, and let . We consider the following two cases.*Case 1. * is -open. Then, is -closed. Since , then . But is -closed. Then, . This shows that .*Case 2. * is -closed. Then, is -open. Since , then . But is -closed. Then, and hence .

Corollary 4.16. *Every -space is -space.*

*Proof. *This is an immediate consequence of Theorems 2.16 and 4.15.

The converse of Corollary 4.16 need not be true, in general, and as a sample, we give the following example.

*Example 4.17. *Let . Define as follows:
Consider the -topology . Then, is -space but not -space.

Theorem 4.18. *Let be a -topological space. Then, the following conditions are equivalent:*(i)* is -space,*(ii)* is and each is -closed.*

*Proof. *Obvious.

#### 5. -Continuous and -Continuous Mappings

*Definition 5.1. *A fuzzy mapping is called(i)fuzzy -continuous if the inverse image of every closed fuzzy set in is -closed fuzzy set in ,(ii)fuzzy -continuous if the inverse image of every closed fuzzy set in is -closed fuzzy set in .

Theorem 5.2. *Every fuzzy -continuous (resp.-continuous) mapping is fuzzy -continuous (resp. -continuous).*

*Proof. *
Obvious.

The converse of the above Theorem may not be true, in general, by the following example.

*Example 5.3. *Let and consider a -topology of Example 4.3, . If is the identity fuzzy mapping, then is fuzzy -continuous but not fuzzy -continuous, since and but . Also, is fuzzy -continuous but not fuzzy -continuous.

Theorem 5.4. *Let be fuzzy mapping and be fuzzy semiregular space. Then, the following conditions are equivalent:*(i)* is fuzzy -continuous,*(ii)* is fuzzy -continuous,*(iii)* is fuzzy -continuous.*

*Proof. *It follows directly from Theorems 2.11 and 4.7(i).

*Definition 5.5. *A fuzzy mapping is called(i)fuzzy -irresolute if the inverse image of every -closed fuzzy set in is -closed fuzzy set in ,(ii)fuzzy -irresolute if the inverse image of every -closed fuzzy set in is -closed fuzzy set in .

Theorem 5.6. *Let and be two fuzzy mappings. Then, *(i)* is fuzzy -continuous if is fuzzy continuous and is fuzzy -continuous,*(ii)* is fuzzy -irresolute if is fuzzy -irresolute and is fuzzy -irresolute, *(iii)* is fuzzy -continuous if is fuzzy -continuous and is fuzzy -irresolute.*

*Proof. *Obvious.

Theorem 5.7. *Let and be two fuzzy mappings. Then, *(i)* is fuzzy -continuous if is fuzzy continuous and is fuzzy -continuous,*(ii)* is fuzzy -irresolute if is fuzzy -irresolute and is fuzzy -irresolute, *(iii)* is fuzzy -continuous if is fuzzy -continuous and is fuzzy -irresolute, *(iv)*Let be -space. Then, is fuzzy -continuous if is fuzzy -continuous and is fuzzy -continuous, *(v)*Let be a fuzzy semiregular space. Then, is fuzzy -continuous if is fuzzy -continuous and is fuzzy -irresolute.*

*Proof. *Obvious.

*Definition 5.8. *A fuzzy mapping is called fuzzy -open if and only if is -open in for any -open fuzzy set in .

Theorem 5.9. *If a fuzzy mapping is bijective, fuzzy -open, and fuzzy -continuous, then is fuzzy -irresolute.*

*Proof. *Let be -closed fuzzy set in , and let , where . Clearly, . Since and is -closed in , then and thus . Since is fuzzy -continuous and is closed in , then is -closed in and hence . Thus, and so is -closed in . This shows that is fuzzy -irresolute.

Theorem 5.10. *If a fuzzy mapping is bijective, fuzzy -open, and fuzzy -continuous, then is fuzzy -irresolute.*

*Proof. *Similar to the proof of Theorem 5.9.

Theorem 5.11. *If a fuzzy mapping is fuzzy -irresolute and is -space, then is fuzzy -continuous.*

*Proof. *Let be a -closed fuzzy set in . By using Theorem 4.2, is -closed. Since is fuzzy -irresolute, then is -closed in . Since is -space, then is -closed in . Thus, is fuzzy -continuous.

Theorem 5.12. *If a mapping is fuzzy -continuous and fuzzy -closed, and is -closed fuzzy set in , then n is -closed in .*

*Proof. *Let be -closed in , and let , where is -open fuzzy set in . Since , is -closed fuzzy set in and since is -open in , then . Thus . Hence, , since is fuzzy -closed. Hence, is -closed in .

Theorem 5.13. *Let be an -space. If a fuzzy mapping be surjective, fuzzy -irresolute, and fuzzy -closed, then is also -space.*

*Proof. *Let be -closed fuzzy set in . Since is fuzzy -irresolute, then is -closed in . Since is -space, then is -closed in . Thus, is -closed in , since is surjective and fuzzy -closed. Hence, is -space.

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