International Journal of Fuzzy and Systems. ISSN 2248-9940 Volume 4, Number 2 (2014), pp. 237-243 © Research India Publications http://www.ripublication.com

A Short Note on Fuzzy -Baire Spaces

G. Thangaraj and E. Poongothai

Department of Mathematics, Thiruvalluvar University Vellore-632 115, Tamilnadu, India Department of Mathematics, Shanmuga Industries Arts and Science College, Tiruvannamalai-606601, Tamilnadu, India.

Abstract

In this paper several characterizations of fuzzy -Baire spaces are studied and the conditions under which a fuzzy becomes a fuzzy - are investigated.

KEY WORDS: Fuzzy -, Fuzzy -set, fuzzy dense, fuzzy nowhere dense, fuzzy -nowhere dense, fuzzy P-space, fuzzy hyperconnected space, fuzzy Baire space.

2000 AMS CLASSIFICATION: 54 A 40, 03 E 72.

INTRODUCTION The fundamental concept of a fuzzy set introduced by Zadeh [11] in 1965, provides a natural foundation for building new branches of fuzzy mathematics. After the discovery of the fuzzy sets, many attempts have been made to extend various branches of mathematics to the fuzzy setting. Fuzzy topological spaces as a very natural generalization of topological spaces were first put forward in the literature by Chang [3] in 1968. Many authors used Chang’s definition in many direction to obtain some results which are compatible with results in general . The concept of-nowhere was introduced and studied by Jiling Cao and Sina Greenwood [4]. The concept of -nowhere dense set in fuzzy setting was introduced by the authors and by means of fuzzy-nowhere dense sets, the concept of fuzzy-Baire spaces was introduced and studied in [9]. In this paper, we investigate several characterizations of fuzzy -Baire spaces and the characterizations of fuzzy - first category sets in fuzzy P-spaces are studied.

Paper Code: 27171 - IJFMS 238 G. Thangaraj and E. Poongothai

PRELIMINARIES By a fuzzy topological space we shall mean a non- X together with a topology T ( in the sense of CHANG ) and denote it by (X,T). Definition 2.1: Let and be any two fuzzy sets in a fuzzy topological space (X,T). Then we define 1. ∨ : → [0,1] as follows: ( ∨ )() = {(), ()}where ∈ , 2. ∧ : → [0,1] as follows: ( ∧ )() = {(), ()}where ∈ , 3. = ⇔ () = 1 − ()where ∈ .

More generally, for a family{ / ∈ } of fuzzy sets in (X,T), the union = ∨ and intersection = ∧ are defined respectively as() = {(), ∈ } and () = {(), ∈ }. Definition 2.2 [1]: Let (X,T) be a fuzzy topological space. For a fuzzy set of X, the and the of are defined respectively as () = ∨ {/ ≤ , ∈ } and () = ∧ {/ ≤ , 1 − ∈ }. Lemma 2.1 [1]: Let be any fuzzy set in a fuzzy topological space (X,T). Then 1 − () = (1 − ) and 1 − () = (1 − ). Definition 2.3 [2]: Let (X,T) be a fuzzy topological space and be a fuzzy set in X. Then is called a fuzzy -set if = ∧ () for each ∈ . Definition 2.4 [2]: Let (X,T) be a fuzzy topological space and be a fuzzy set in X. Then is called a fuzzy -set if = ∨ () for each 1 − ∈ .

Lemma 2.2 [1]: For a family = {}of fuzzy sets of a fuzzy space X, ∨ () ≤ ∨ (). In case A is a ,∨ () = ∨ (). Also ∨ () ≤ ∨ (). Definition 2.5 [6]: A fuzzy set in a fuzzy topological space (X,T) is called fuzzy dense if there exists no fuzzy in (X,T) such that < < 1. That is., () = 1. Definition 2.6 [7]: A fuzzy set in a fuzzy topological space (X,T) is called fuzzy nowhere dense if there exists no non-zero fuzzy in (X,T) such that < (). That is., () = 0. Definition 2.7 [9]: A fuzzy set in a fuzzy topological space (X,T) is called fuzzy - nowhere dense if is a fuzzy -set in (X,T) such that () = 0. Definition 2.8 [9]: A fuzzy set in a fuzzy topological space (X,T) is called a fuzzy-first category set if = ∨ (),where ’s are fuzzy -nowhere dense sets in (X,T). Any other fuzzy set in (X,T) is said to be of fuzzy -second category. Definition 2.9 [9]: Let be a fuzzy -first category set in a fuzzy topological space (X,T). Then, the fuzzy set 1 − is called a fuzzy -residual set in (X,T). Definition 2.10 [10]: A fuzzy topological space (X,T) is called afuzzy P-space, if

A Short Note on Fuzzy σ-Baire Spaces 239 countable intersection of fuzzy open sets in (X,T) is fuzzy open. That is, every non- zero fuzzy -set in (X,T) is fuzzy open in (X,T). Definition 2.11 [5]: A fuzzy topological space (X,T) is called a fuzzy hyperconnected space if every non-null fuzzy open set is fuzzy dense in (X,T).

FUZZY -BAIRE SPACES Definition 3.1 [9]: Let (X,T) be a fuzzy topological space. Then (X,T) is called a fuzzy -Baire space if ∨ () = 0, where the fuzzy sets ()’s are fuzzy - nowhere dense sets in (X,T). Theorem 3.1 [9]: Let (X,T) be a fuzzy topological space. Then the following are equivalent: 1. (X,T) is a fuzzy -Baire space. 2. () = 0 for every fuzzy -first category set in (X,T). 3. () = 1 for every fuzzy -residual set in (X,T). The following proposition ensures the existence of fuzzy first category sets in a fuzzy topological space if there are fuzzy -nowhere dense sets in (X,T). Proposition 3.1: If is a fuzzy -nowhere dense set in a fuzzy topological space (X,T),then is a fuzzy first category set in (X,T). Proof: Let be a fuzzy -nowhere dense set in a fuzzy topological space (X,T). Thenis a fuzzy fuzzy-set such that () = 0. Since is a fuzzy -set, = ∨ (), where’s are fuzzy closed sets in (X,T). Now () = 0 implies that ∨ () = 0. But, from lemma 2.2, we have ∨ () ≤ ∨ (). This implies that∨ () ≤ 0. That is, ∨ () = 0. Hence we have () = 0 for each i. Since()’s are fuzzy closed, () = .Then () = () = 0. That is, () = 0 for each i. Hence ()’s are fuzzy nowhere dense sets in (X,T). Therefore = ∨ (), where ()’s are fuzzy nowhere dense sets, implies that is a fuzzy first category set in (X,T).

The existence of fuzzy nowhere dense sets which are fuzzy , ensures the existence of fuzzy -nowhere dense sets in a fuzzy topological space.

Proposition 3.2: If the fuzzy nowhere dense set is a fuzzy -set in a fuzzy topological space (X,T), then is a fuzzy -nowhere dense set in (X,T).

Proof: Let the fuzzy nowhere dense set be a fuzzy -set in (X,T). Then = ∨ (), where1 − ∈ . Since is a fuzzy nowhere dense set, we have () = 0 in (X,T). Now () ≤ () implies that () = 0. Hence is fuzzy -set such that () = 0.Then is a fuzzy -nowhere dense set in (X,T).

Proposition 3.3: If each fuzzy nowhere dense set is a fuzzy -set in a fuzzy topological space (X,T), then is a fuzzy first category set in (X,T).

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Proof: Let each fuzzy nowhere dense set be a fuzzy -set in (X,T). Since is a fuzzy nowhere dense set in (X,T), we have () = 0. Now () ≤ (), implies that () ≤ 0. That is., () = 0. Hence is a fuzzy -set such that () = 0.Then is a fuzzy -nowhere dense set in (X,T). Then, by proposition 3.1, is a fuzzy first category set in (X,T). Proposition 3.4: If a fuzzy -first category set is a fuzzy closed set in a fuzzy - Baire space (X,T), then is a fuzzy nowhere dense set in (X,T). Proof: Let be a fuzzy -first category set in (X,T). Since (X,T) is a fuzzy -Baire space by theorem 3.1, () = 0. By hypothesis, is fuzzy closed in (X,T). This implies that () = . Then () = () = 0. That is, () = 0 in (X,T). Hence (X,T) is a fuzzy nowhere dense set in (X,T). Proposition 3.5: If a fuzzy -first category set in fuzzy topological space (X,T) is a fuzzy nowhere dense set, then (X,T) is a fuzzy -Baire space. Proof: Let be a fuzzy -first category set in a fuzzy topological space (X,T) such that () = 0. Then () ≤ () implies that () = 0.Hence, for a fuzzy -first category set in (X,T), we have () = 0. Hence by theorem 3.1, (X,T) is a fuzzy -Baire space.

Theorem 3.2 [8]: If is a fuzzy first category set in (X,T), then there is a fuzzy -set in (X,T) such that ≤ . Proposition 3.6: If is a fuzzy -nowhere dense set in a fuzzy topological space (X,T), then there exists a fuzzy -set in (X,T) such that ≤ . Proof: Let be a fuzzy -nowhere dense set in a fuzzy topological space (X,T). Then by proposition 3.1, is a fuzzy first category set in (X,T). Then by theorem 3.2, there is a fuzzy -set in (X,T) such that ≤ . Proposition 3.7: If (X,T) is a fuzzy -first category space, then ∨ () = 1, where ()’s are fuzzy closed sets in (X,T). Proof: Let (X,T) be a fuzzy -first category space. Then ∨ () = 1…….(1), where ()’s are fuzzy -nowhere dense sets in (X,T). Since ()’s ( i= 1 to ∞) are fuzzy -nowhere dense sets in (X,T), by proposition 3.6, there exists -sets in (X,T) such that ≤ . Since ()’s ( i= 1 to ∞) are fuzzy -sets, = ∨ , where ’s are fuzzy closed sets in (X,T). Now ≤ implies that ∨ () ≤ ∨ () = ∨∨ . Let ∨∨ = ∨ (), where = and ()’s are fuzzy closed sets in (X,T). Hence ∨ () ≤ ∨ (). Then from (1), 1 ≤ ∨ (). Hence we have ∨ () = 1,where ()’s are fuzzy closed sets in (X,T).

Proposition 3.8: If ()’s ( i= 1 to ∞) are fuzzy nowhere dense sets such that ≤ , where ()’s are fuzzy -sets in a fuzzy Baire space (X,T), then (X,T) is a fuzzy - Baire space.

Proof: Let ( i= 1 to ∞) be a fuzzy nowhere dense set in a fuzzy Baire space (X,T)

A Short Note on Fuzzy σ-Baire Spaces 241

such that ≤ , where is a fuzzy -set in (X,T). Then () ≤ ()…….(1). Since is a fuzzy nowhere dense set, () = 0. Now () ≤ () implies that () = 0. Hence we have from (1) () ≤ 0. That is, () = 0 and hence () is a fuzzy -set such that () = 0. This implies that is a fuzzy -nowhere dense set in (X,T). Now ≤ implies that ∨ () ≤∨ () and hence ∨ () ≤ ∨ (). Since (X,T) is a fuzzy Baire space ∨ () = 0. This implies that ∨ () ≤ 0. That is, ∨ () = 0, where ()’s are fuzzy-nowhere dense sets in (X,T). Therefore, (X,T) is a fuzzy -Baire space.

FUZZY -FIRST CATEGORY SETS AND FUZZY P-SPACES Proposition 4.1: If is a fuzzy −first category set in a fuzzy P-space (X,T), then is not a fuzzy dense set in (X,T). Proof: Let us assume the contrary. Suppose that is a fuzzy -first category set in (X,T) such that () = 1. Then = ∨ (), where ()’s are fuzzy -nowhere dense sets in (X,T). Since ()’s are fuzzy-nowhere dense sets, ()’s are fuzzy- sets and () = 0 ( i= 1 to ∞). Then (1 − )’s are fuzzy -sets in (X,T). Since (X,T) is a fuzzy P-space, the fuzzy -sets (1 − )’s are fuzzy open sets in (X,T). Let = ∧ (1 − ). Then is a non-zero fuzzy -set in (X,T). Since (X,T) is a fuzzy P-space, is a fuzzy open set in (X,T) and hence () = . Now = 1 − ∨ () = 1 − . Then we have () = (1 − ) = 1 − () = 1 − 1 = 0, a contradiction. Hence, in a fuzzy P-space (X,T), fuzzy -first category sets are not fuzzy dense sets in (X,T). Proposition 4.2: If is a fuzzy -first category set in a fuzzy P-space then (1 − ) ≠ 0. Proof: Let be a fuzzy -first category set in a fuzzy P-space (X,T). Then by proposition 4.1, is not a fuzzy dense set. That is,() ≠ 1. Then, 1 − () ≠ 0. This implies that (1 − ) ≠ 0. Remark: In view of the above proposition 4.2, we have the following: In a fuzzy P- space (X,T), the interior of a fuzzy -residual set is non-zero in (X,T). Proposition 4.3: If is a fuzzy -first category set in a fuzzy P-space (X,T), then is a fuzzy -set in (X,T). Proof: Let be a fuzzy -first category set in (X,T). Then = ∨ (), where ()’s are fuzzy -nowhere dense sets in (X,T). Since ()’s are fuzzy -nowhere dense sets, ()’s are fuzzy -sets and () = 0 ( i= 1 to ∞). Then (1 − )’s are fuzzy -sets in (X,T). Since (X,T) is a fuzzy P-space, the fuzzy -sets (1 − )’s are fuzzy open sets in (X,T). Let = ∧ (1 − ). Then is a non-zero fuzzy -set in (X,T). Then, from = 1 − ∨ (), we have = 1 − and hence = 1 − . Since is a fuzzy -set in (X,T), is a fuzzy -set in (X,T). Hence in a fuzzy P- space (X,T) fuzzy -first category sets are fuzzy -sets in (X,T).

242 G. Thangaraj and E. Poongothai

If and are any two fuzzy-nowhere dense sets in a fuzzy topological space (X,T), then ∨ need not be a fuzzy -nowhere dense set in (X,T). If()’s ( i= 1 to ∞) are fuzzy -nowhere dense sets in a fuzzy topological space (X,T), when ∨ () is a fuzzy -nowhere dense set in (X,T) ? The answer for this question is obtained in the following proposition. Proposition 4.4: If (X,T) is a fuzzy-Baire and fuzzy P-space, then each fuzzy -first category set is a fuzzy -nowhere dense set in (X,T). Proof: Let be a fuzzy -first category set in a fuzzy -Baire and fuzzy P-space (X,T). Then = ∨ (), where ()’s are fuzzy -nowhere dense sets in (X,T). Since (X,T) is a fuzzy −Baire space, by theorem 3.1, () = 0. Since (X,T) is a fuzzy P-space, by proposition 4.3, the fuzzy -first category set is a fuzzy -set in (X,T). Hence is a fuzzy -set such that () = 0. Then is a fuzzy-nowhere dense set in (X,T). Hence, if (X,T) is a fuzzy-Baire and fuzzy P-space, then each fuzzy -first category set is a fuzzy -nowhere dense set in (X,T). Proposition 4.5: If is a fuzzy -first category set in a fuzzy hyperconnected and fuzzy P-space (X,T), then is a fuzzy -nowhere dense set in (X,T). Proof: Let be a fuzzy -first category set in a fuzzy hyperconnected and fuzzy P- space (X,T). Since (X,T) is a fuzzy P-space, by proposition 4.3, is a fuzzy -set and hence1 − is a fuzzy -set. Since (X,T) is a fuzzy P-space, 1 − is fuzzy open in (X,T). Again since (X,T) is a fuzzy hyperconnected space the fuzzy open set1 − is fuzzy dense in (X,T). That is (1 − ) = 1. Then we have () = 0. Hence is a fuzzy-set such that () = 0. Therefore, in a fuzzy hyperconnected and fuzzy P-space (X,T), the fuzzy -first category set is a fuzzy -nowhere dense set in (X,T). Proposition 4.6: If is a fuzzy -first category set in a fuzzy hyperconnected and fuzzy P-space (X,T), then is a fuzzy first category set in (X,T). Proof: Let be a fuzzy-first category set in a fuzzy hyperconnected and fuzzy P- space (X,T). Then by proposition 4.5, is a fuzzy -nowhere dense set in (X,T). Then by proposition 3.1, is a fuzzy first category set in (X,T). Proposition 4.7: If (X,T) isa fuzzy Baire, fuzzy P-spaceand fuzzy hyperconnected space, then (X,T) is a fuzzy -Baire space. Proof: Let be a fuzzy-first category set in a fuzzy Baire, fuzzy P-space and fuzzy hyperconnected space (X,T). Then by proposition 4.6, is a fuzzy first category set in (X,T). Since (X,T) is a fuzzy Baire space, by theorem 3.1, () = 0. Hence, for the fuzzy -first category set in (X,T), we have () = 0. Then, by theorem 3.1, (X,T) is a fuzzy -Baire space. Proposition 4.8: If each fuzzy open set is fuzzy dense in a fuzzy Baire and fuzzy P- space (X,T), then (X,T) is a fuzzy -Baire space. Proof: Let be a fuzzy -first category set in a fuzzy Baire and fuzzy P-space (X,T). Since (X,T) is a fuzzy P-space by proposition 4.3, the fuzzy -first category set is a

A Short Note on Fuzzy σ-Baire Spaces 243

fuzzy -set in (X,T). Then 1 − is a fuzzy -set in (X,T). Again, since (X,T) is a fuzzy P-space, 1 − is a fuzzy open in (X,T). By hypothesis, 1 − is fuzzy dense in (X,T). That is,(1 − ) = 1. Then 1 − () = 1 and hence () = 0.Hence, for fuzzy -first category set in (X,T), we have () = 0.Then, by theorem 3.1, (X,T) is a fuzzy -Baire space. Proposition 4.9: If is a fuzzy -nowhere dense set in a fuzzy P-space (X,T), then is not a fuzzy dense set in (X,T). Proof: Let be a fuzzy -nowhere dense set in a fuzzy P-space (X,T). Then by proposition 3.6, there exists a fuzzy-set in (X,T) such that ≤ . Then 1 − ≥ 1 − . Since is a fuzzy -set, 1 − is a fuzzy-set in (X,T). Since (X,T) is a fuzzy P-space, 1 − is a fuzzy open set in (X,T). That is, (1 − ) = 1 − … … . (1). Now ≤ implies that 1 − ≥ 1 − . Then (1 − ) ≥ (1 − ) and hence (1 − ) ≥ 1 − ( from (1) ). Since 1 − ≠ 0, (1 − ) ≠ 0. Then 1 − () ≠ 0. This implies that() ≠ 1. Hence, if is a fuzzy -nowhere dense set in a fuzzy P-space, then () ≠ 1. Therefore, is not a fuzzy dense set in (X,T).

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