ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II)

Transcript

1 Commun. Korean Math. Soc. 25 (2010), No. 3, pp DOI /CKMS ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) Ahmed Abd El-Kader Ramadan, Salah El-Deen Abbas, and Ahmed Aref Abd El-latif Abstract. In this paper, we have use a fuzzy bitopological space (X, τ 1, τ 2 ) to create a family τ s which is a supra fuzzy topology on X. Also, we introduce and study the concepts of r (τ i, τ )-generalized fuzzy regular closed, r (τ i, τ )-generalized fuzzy strongly semi-closed and r (τ i, τ )- generalized fuzzy regular strongly semi-closed sets in fuzzy bitopological space in the sense of Šostak. Also, these classes of fuzzy subsets are applied for constructing several type of fuzzy closed mapping and some type of fuzzy separation axioms called fuzzy binormal, fuzzy mildly binormal and fuzzy almost pairwise normal. 1. Introduction and preliminaries The concept of fuzzy topology was first defined in 1968 by Chang [2]. A Chang s fuzzy topology is a crisp subfamily of some family of fuzzy sets and fuzziness in the concept of openness of a fuzzy set has not been considered, which seems to be a drawback in the process of fuzzification of the concept of the topological space. Therefore, in 1985 Šostak [18], introduce the fundamental concept of a fuzzy topological structure as an extension of both crisp topology and Chang s fuzzy topology, in the sense that not only the obect were fuzzified, but also the axiomatics. In [19, 20] Šostak gave some rules and showed how such an extension can be realized. Chattopadhyay et. al [3, 4] have redefined the similar concept. In [15, 5] Ramadan gave a similar definition namely Smooth fuzzy topology for lattice L = [0, 1], it has been developed in many direction [7, 9, 10, 12, 13, 17]. In this paper, we introduce and study the concepts of r (τ i, τ )-generalized fuzzy regular closed, r (τ i, τ )-generalized fuzzy strongly semi-closed and r (τ i, τ )-generalized fuzzy regular strongly semi-closed sets which are based on the alternative effect of fuzzy closure and fuzzy interior operators with respect to two fuzzy topologies. Also, these classes of fuzzy subsets are applied for constructing several types of fuzzy closed mappings Received June 8, 2006; Revised February 8, Mathematics Subect Classification. 54A40. Key words and phrases. fuzzy bitopological space, supra fuzzy topology, fuzzy binormal, fuzzy mildly binormal, fuzzy almost pairwise normal. 457 c 2010 The Korean Mathematical Society

2 458 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF and some types of fuzzy separation axioms called fuzzy binormal, fuzzy mildly binormal and fuzzy almost pairwise normal. Throughout this paper, let X be a nonempty set I = [0, 1], I 0 = (0, 1] and I X denote the set of all fuzzy subset of X. For α I, α = α for every x X. A fuzzy set λ is quasi-coincident with a fuzzy set µ, denoted by λqµ, if there exists x X such that λ(x) + µ(x) > 1, if λ is not quasi-coincident with µ, we denote λ qµ, λ qµ if and only if λ 1 µ [14]. Definition 1.1 ([6, 18]). A mapping τ : I X I is called a supra fuzzy topology on X if it satisfies the following conditions: (S1) τ(0) = τ(1) = 1. (S2) τ( i J µ i) i J τ(µ i) for any {µ i : i J} I X. The pair (X, τ) is called supra fuzzy topological space (briefly, sfts). A supra fuzzy topology τ is called fuzzy topology on X if (T) τ(µ 1 µ 2 ) τ(µ 1 ) τ(µ 2 ) for any µ 1, µ 2 I X. The pair (X, τ) is called fuzzy topological space (briefly fts). The triple (X, τ 1, τ 2 ) is called fuzzy bitopological space (briefly, fbts), where τ 1 and τ 1 are fuzzy topologies on X. Throughout this paper, the indices i, {1, 2} and i. Theorem 1.1 ([4]). Let (X, τ) be a fts. For each λ I X and r I 0 we define an operator C τ : I X I 0 I X as follows: C τ (λ, r) = {µ : λ µ, τ(1 µ) r}. For each λ, µ I X and r, s I 0 the operator atisfies the following conditions: (i) C τ (0, r) = 0. (ii) λ C τ (λ, r). (iii) C τ (λ, r) C τ (µ, r) = C τ (λ µ, r). (iv) C τ (λ, r) C τ (λ, s) if r s. (v) C τ (C τ (λ, r), r) = C τ (λ, r). Theorem 1.2 ([8]). Let (X, τ) be a fts. For each λ I X and r I 0, we define an operator I τ : I X I 0 I X as follows: I τ (λ, r) = {µ : µ λ, τ(µ) r}. For each λ, µ I X and r, s I 0 the operator I τ satisfies the following conditions: (i) I τ (1 λ, r) = 1 C τ (λ, r) and C τ (1 λ, r) = 1 I τ (λ, r). (ii) I τ (1, r) = 1. (iii) I τ (λ, r) λ. (iv) I τ (λ, r) I τ (µ, r) = I τ (λ µ, r). (v) I τ (λ, r) I τ (λ, s) if r s. (vi) I τ (I τ (λ, r), r) = I τ (λ, r). Definition 1.2 ([16, 11]). Let (X, τ 1, τ 2 ) be a fbts. For λ I X and r I 0 :

3 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 459 (i) λ is called r (τ i, τ )-fuzzy strongly semi-open (briefly, r (τ i, τ )-fsso) if there exists µ I X with τ i (µ) r such that µ λ I τi (C τ (µ, r), r). (ii) λ is called r (τ i, τ )-fuzzy strongly semi-closed (briefly, r (τ i, τ )-fssc) if there exists µ I X with τ i (1 µ) r such that C τi (I τ (µ, r), r) λ µ. (iii) λ is called r (τ i, τ )-fuzzy regular open (briefly, r (τ i, τ )-fro) if λ = I τi (C τ (λ, r), r). (iv) λ is called r (τ i, τ )-fuzzy regular closed (briefly, r (τ i, τ )-frc) if λ = C τi (I τ (λ, r), r). The set of all r (τ i, τ )-fsso, r (τ i, τ )-fssc, r (τ i, τ )-fro and r (τ i, τ )-frc sets of a fbts (X, τ 1, τ 2 ) will be denoted by r (τ i, τ ) F SSO(X), r (τ i, τ ) F SSC(X), r (τ i, τ ) F RO(X) and r (τ i, τ ) F RC(X) respectively. Theorem 1.3 ([16]). Let (X, τ 1, τ 2 ) be a fbts. For λ I X and r I 0, the following statements are equivalent: (i) λ is r (τ i, τ )-fsso. (ii) 1 λ is r (τ i, τ )-fssc. (iii) λ I τi (C τ (I τi (λ, r), r), r). (iv) C τi (I τ (C τi (1 λ, r), r), r) 1 λ. Definition 1.3 ([16]). Let (X, τ 1, τ 2 ) be a fbts. For λ I X and r I 0 : (i) The r (τ i, τ )-fuzzy strongly semi-interior of λ, denoted by SSI (λ, r) is defined by SSI (λ, r) = {µ I X : µ λ, µ is r (τ i, τ ) fsso}. (ii) The r (τ i, τ )-fuzzy strongly semi-closure of λ, denoted by SSC (λ, r) is defined by SSC (λ, r) = {µ I X : µ λ, µ is r (τ i, τ ) fssc}. Theorem 1.4 ([16]). Let (X, τ 1, τ 2 ) be a fbts. For λ, µ I X and r I 0 : (i) SSC (0, r) = 0 and SSI (1, r) = 1. (ii) I τi (λ, r) SSI (λ, r) λ SSC (λ, r) C τi (λ, r). (iii) λ µ SSI (λ, r) SSI (µ, r) and SSC (λ, r) SSC (µ, r). (iv) λ is r (τ i, τ )-fsso if and only if λ = SSI (λ, r). (v) λ is r (τ i, τ )-fssc if and only if λ = SSC (λ, r). (vi) SSC (SSC (λ, r), r) = SSC (λ, r). Definition 1.4 ([8]). A mapping f : (X, τ 1, τ 2 ) (X, τ 1, τ 2 ) from a fbts (X, τ 1, τ 2 ) to another fbts (X, τ 1, τ 2 ) is said to be fuzzy pairwise continuous (fpc, for short) if and only if τ i (f 1 (µ)) τ i (µ) for each µ IY and i = 1, 2. If we put τ = τ 1 = τ 2 and τ = τ1 = τ2, then the definition of fuzzy pairwise continuous mapping reduce to the corresponding fuzzy continuous mapping due to Šostak [18].

4 460 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF Definition 1.5 ([1]). A mapping C : I X I 0 I X is called a supra fuzzy closure operator on X if for λ, µ I X and r, s I 0, it satisfies the following conditions: (C1) C(0, r) = 0. (C2) λ C(λ, r). (C3) C(λ, r) C(µ, r) C(λ µ, r). (C4) C(λ, r) C(λ, s) if r s. (C5) C(C(λ, r), r) = C(λ, r). The pair (X, C) is called a supra fuzzy closure space. 2. Some types of fuzzy closeness in fuzzy bitopological spaces Theorem 2.1. Let (X, τ 1, τ 2 ) be a fbts. Define a mapping τ s : IX I on X by: τ s (λ) = {r I 0 : SSI (λ, r) = λ}. Then τ s is a supra fuzzy topology. Proof. (S1) It is clearly since for all r I 0, SSI (0, r) = 0 and SSI (1, r) = 1. (S2) Suppose that there exists λ = k J λ k I X such that There exists r 0 I 0 such that τ s (λ) < k J τ s (λ k ). τ(λ) s < r 0 τ(λ s k ). k J This implies that for all k J, there exists r k J with SSI (λ k, r k ) = λ k such that τ(λ s k ) r k r 0. Then SSI (λ k, r 0 ) SSI (λ k, r k ) = λ k. But by Theorem 1.4(ii), we have SSI (λ k, r 0 ) λ k, thus This implies that SSI (λ, r 0 ) = SSI ( k J By using Theorem 1.4(ii), we obtain SSI (λ k, r 0 ) = λ k. λ k, r 0 ) k J λ = SSI (λ, r 0 ). That is τ s (λ) r 0. It is a contradiction. Thus SSI (λ k, r 0 ) = k J λ k = λ. τ( s λ k ) λ k for any {λ k : λ k I X }. k J k J

5 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 461 Hence, τ s is a supra fuzzy topology. The following example shows that τ s is not fuzzy topology in general. Example 2.1. Let X = {a, b}. Define λ 1, λ 2, λ 3, λ 4 I X as follows: λ 1 (a) = 0.7 λ 1 (b) = 0.1 λ 2 (a) = 0.1 λ 2 (b) = 0.7 λ 3 (a) = 0.7 λ 3 (b) = 0.5 λ 4 (a) = 0.5 λ 4 (b) = 0.7 Define the fuzzy topologies τ 1, τ 2 : I X I as follows: 1, if λ = 0, 1 0.3, if λ = λ 1, λ 2 τ 1 (λ) = τ 2 (λ) = 0.5, if λ = 0.2, , if λ = 0.1, 0.7 0, otherwise. 0, otherwise, Then for 0 < r 0.3, λ 3 and λ 4 are r (τ 1, τ 2 )-fsso sets and λ 3 λ 4 is not r (τ 1, τ 2 )-fsso set which implies that τ s 12(λ 3 ) τ s 12(λ 4 ) > τ s 12(λ 1 λ 3 ) = 0. Then τ s 12 is not fuzzy topology. Theorem 2.2. Let (X, τ 1, τ 2 ) be a fbts. For each λ I X and r I 0 we define a mapping : I X I 0 I X as follows: (λ, r) = {µ : µ λ, τ s (1 µ) r}. Then (X, ) is a supra fuzzy closure space. Proof. (C1), (C2) and (C4) follows directly from the definition of. (C3) Since λ λ µ and µ λ µ we have (λ, r) (λ µ, r) and (µ, r) (λ µ, r). Then (C5) From the definition of (λ, r) (µ, r) (λ µ, r). we obtain (1) (λ, r) ( (λ, r), r) for each λ I X, r r 0. Conversely, suppose that there exist λ I X, r I 0 and x X such that ( (λ, r), r)(x) > (λ, r)(x). By the definition of (λ, r), there exists µ I X with µ λ and such that SSI (1 µ, r) = 1 µ ( (λ, r), r)(x) > µ(x) (λ, r)(x). On the other hand, since µ λ and τ s (1 µ) r, then (λ, r) (µ, r) = µ

6 462 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF therefore It is a contradiction. Thus ( (λ, r), r) (µ, r) = µ. (2) ( (λ, r), r) (λ, r) for each λ I X, r r 0. From (1) and (2) we obtain (C5). Theorem 2.3. Let (X, τ 1, τ 2 ) be a fbts. For λ I X and r r 0 we define a mapping I τ s : I X I 0 I X as follows: Then we have I τ s (λ, r) = {µ : µ λ, τ s (µ) r}. I τ s (1 λ, r) = 1 (λ, r). Proof. For λ I X, r I 0 we have 1 (λ, r) = 1 {µ : µ λ, τ s (1 µ) r} = {1 µ : 1 µ 1 λ, τ s (1 µ) r} = I τ s (1 λ, r). Definition 2.1. Let (X, τ 1, τ 2 ) be a fbts and λ I X. Then, for r r 0, λ is called: (i) r (τ i, τ )-generalized fuzzy closed (briefly, r (τ i, τ )-gfc) set if λ µ and τ i (µ) r implies C τ (λ, r) µ. (ii) r (τ i, τ )-generalized fuzzy regular closed (briefly, r (τ i, τ )-gfrc) set if λ µ and µ is r (τ i, τ )-fro set implies C τ (λ, r) µ. (iii) r (τ i, τ )-generalized fuzzy strongly semi-closed (briefly, r (τ i, τ )- gfssc) set if λ µ and τ i (µ) r implies (λ, r) µ. (iv) r (τ i, τ )-generalized fuzzy regular strongly semi-closed (briefly, r (τ i, τ )-gfrssc) set if λ µ and µ is r (τ i, τ )-fro set implies (λ, r) µ. The set of all r (τ i, τ )-gfc, r (τ i, τ )-gfrc, r (τ i, τ )-gfssc and r (τ i, τ )- gfrssc sets of a fbts (X, τ 1, τ 2 ) will be denoted by r (τ i, τ ) GF C(X), r (τ i, τ ) GF RC(X), r (τ i, τ ) GF SSC(X) and r (τ i, τ ) GF RSSC(X) respectively. Remark 2.1. The following diagram shows the relation between the above different types of fuzzy closeness in a fbts (X, τ 1, τ 2 ). r (τ i, τ )-fssc r (τ i, τ )-gfssc r (τ i, τ )-gfc r (τ i, τ )-gfrssc r (τ i, τ )-gfrc

7 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 463 Example 2.2. Let X = {a, b, c}. We define the fuzzy topologies τ 1, τ 2 : I X I as follows: 1, if λ = 0, 1 1 τ 1 (λ) = 3, if λ = χ {a} 1 2, if λ {χ {a,b}, χ {a,c} } 0, otherwise 1 τ 2 (λ) = 3, if λ {χ {c}, χ {a,c} } 0, otherwise. For 0 < r 1 3, χ {a} is r (τ 1, τ 2 )-gfrc but not r (τ 1, τ 2 )-gfc. Example 2.3. Let X = {a, b, c, d}. We define the fuzzy topologies τ 1, τ 2 : I X I as follows: 1 τ 1 (λ) = 2, if λ {χ {d}, χ {a,d} } 0, otherwise 1, if λ = 0, 1 1 τ 2 (λ) = 2, if λ {χ {c}, χ {a,b} } 3 4, if λ {χ {c,d}, χ {a,b,c} } 0, otherwise. For 0 < r 1 2, χ {a} is r (τ 1, τ 2 )-gfrssc but not r (τ 1, τ 2 )-gfrc. Also, χ {a} is r (τ 1, τ 2 )-gfssc but not r (τ 1, τ 2 )-gfc, χ {a,c} is r (τ 2, τ 1 )-gfssc but not r (τ 2, τ 1 )-fssc and χ {c,d} is r (τ 2, τ 1 )-gfrssc but not r (τ 2, τ 1 )-gfssc. Remark 2.2. The complement of r (τ i, τ )-gfc (resp. r (τ i, τ )-gfrc, r (τ i, τ )- gfssc, r (τ i, τ )-gfrssc) set is called r (τ i, τ )-gfo (resp. r (τ i, τ )-gfro, r (τ i, τ )-gfsso, r (τ i, τ )-gfrsso) set. r (τ i, τ )-fsso r (τ i, τ )-gfsso r (τ i, τ )-gfo r (τ i, τ )-gfrsso r (τ i, τ )-gfro The set of all r (τ i, τ )-gfo, r (τ i, τ )-gfro, r (τ i, τ )-gfsso and r (τ i, τ )- gfrsso sets of a fbts (X, τ 1, τ 2 ) will be denoted by r (τ i, τ ) GF O(X), r (τ i, τ ) GF RO(X), r (τ i, τ ) GF SSO(X) and r (τ i, τ ) GF RSSO(X) respectively. Lemma 2.1. Let (X, τ 1, τ 2 ) be a fbts and λ I X. For r r 0 λ is:

8 464 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF (i) r (τ i, τ )-gfo set if µ λ and τ i (1 µ) r, then µ I τ (λ, r). (ii) r (τ i, τ )-gfro set if µ λ and µ is r (τ i, τ )-frc set, then µ I τ (λ, r). (iii) r (τ i, τ )-gfsso set if µ λ and τ i (1 µ) r, then µ I τ s (λ, r). (iv) r (τ i, τ )-gfrsso set if µ λ and µ is r (τ i, τ )-frc set, then µ I τ s (λ, r). Theorem 2.4. Let λ be r (τ i, τ )-fsso set in a fbts (X, τ 1, τ 2 ). Then C τ (λ, r) = C τ (I τi (λ, r), r) = (λ, r). Proof. First: Let λ r (τ i, τ ) F SSO(X). λ I τi (C τ (I τi (λ, r), r), r). This implies that By Theorem 1.3, we have C τ (λ, r) C τ (I τi (C τ (I τi (λ, r), r), r), r) = C τ (I τi (λ, r), r). Conversely, since I τi (λ, r) λ C τ (λ, r), then C τ (I τi (λ, r), r) C τ (λ, r). Hence, C τ (λ, r) = C τ (I τi (λ, r), r). Second: Since λ is r (τ i, τ )-fsso set, then Therefore λ I τi (C τ (I τi (λ, r), r), r) C τ (I τi (λ, r), r). (λ, r) (C τ (I τi (λ, r), r), r) = C τ (I τi (λ, r), r) = C τ (λ, r). Conversely, suppose that (λ, r) < C τ (λ, r). Then, there exist ρ r (τ, τ i ) F SSC(X) and x X such that λ ρ and (λ, r)(x) ρ(x) < C τ (λ, r)(x). Since λ ρ, then C τ (λ, r) C τ (ρ, r). Since λ r (τ, τ i ) F SSO(X), then by the first part we have, C τ (I τi (λ, r), r) = C τ (λ, r) C τ (ρ, r). Then Thus λ I τi (C τ (I τi (λ, r), r), r) I τi (C τ (ρ, r), r). C τ (λ, r) C τ (I τi (C τ (ρ, r), r), r) ρ. It is a contradiction. Then C τ (λ, r) (λ, r). Thus C τ (λ, r) = (λ, r). From the first and second parts we have, C τ (λ, r)=c τ (I τi (λ, r), r)= (λ, r). 3. Some types of fuzzy closed mappings in fuzzy bitopological spaces Definition 3.1. Let f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) be a mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ 1, τ 2 ). Then f is called: (i) -fuzzy almost strongly semi-closed (briefly, -fass closed) mapping if f(λ) r (τ i, τ ) F SSC(Y ) for each λ r (τ i, τ ) F RC(X). (ii) -fuzzy almost generalized strongly semi-closed (briefly, -fagss closed) mapping if f(λ) r (τ i, τ ) GF SSC(Y ) for each λ r (τ, τ i ) F RC(X).

9 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 465 (iii) -fuzzy almost generalized regular strongly semi-closed (briefly, fagrss closed) mapping if f(λ) r (τi, τ ) GF RSSC(Y ) for each λ r (τ, τ i ) F RC(X). (iv) -fuzzy closed if and only if τ (1 f(λ)) τ (1 λ) for each λ I X. Remark 3.1. From the above definition the implications contained in the following diagram are true τ -fuzzy closed -fass closed -fagss closed -fagrss closed. The following examples, show that the reverse may not be true in general. Example 3.1. Let X = {a, b, c}. Define the fuzzy topologies τ 1, τ 2, τ1, τ2 : I X I as follows: 1, if λ = 0, 1 0.3, if λ {χ {a}, χ {b} } τ 1 (λ) = 0.5, if λ = χ {a,b} 0, otherwise τ 2 (λ) = 0.5, if λ {χ {c}, χ {b,c} } 0, otherwise τ1 (λ) = 0.4, if λ {χ {b}, χ {a,c} } 0, otherwise 1, if λ = 0, 1 τ2 0.4, if λ {χ {a}, χ {c} } (λ) = 0.5, if λ = χ {a,c} 0, otherwise. Then for 0 < r 0.3 the identity mapping f : (X, τ 1, τ 2 ) (X, τ 1, τ 2 ) is 21-fagrss closed but not 21-fagss closed, since χ {c} r (τ 1, τ 2 ) F RC(X) but f(χ {c} ) r (τ 2, τ 1 ) GF SSC(X). Example 3.2. Let X = {a, b, c}. Define the fuzzy topologies τ 1, τ 2, τ1,τ2 : I X I as follows: τ 1 (λ) = 0.7, if λ {χ {a}, χ {a,b} } 0, otherwise τ 2 (λ) = 0.5, if λ {χ {b}, χ {b,c} } 0, otherwise

10 466 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF τ1 (λ) = 0.4, if λ = χ {c} 0, otherwise τ2 (λ) = 0.6, if λ = χ {a,c} 0, otherwise. Then for 0 < r 0.4 the identity mapping f : (X, τ 1, τ 2 ) (X, τ 1, τ 2 ) is 21-fagss closed but not 12-fass closed, since χ {b,c} r (τ 1, τ 2 ) F RC(X) but f(χ {b,c} ) r (τ 1, τ 2 ) F SSC(X). Example 3.3. Let X = {a, b, c}. Define λ 1, λ 2, λ 3, λ 4 I X as follows: λ 1 (a) = 0.8 λ 1 (b) = 0.7 λ 1 (c) = 0.6 λ 2 (a) = 1.0 λ 2 (b) = 0.8 λ 2 (c) = 0.7 λ 3 (a) = 0.3 λ 3 (b) = 0.4 λ 3 (c) = 0.5 λ 4 (a) = 0.3 λ 4 (b) = 0.5 λ 4 (c) = 0.6 Define the fuzzy topologies τ 1, τ 2, τ1, τ2 : I X I as follows: τ 1 (λ) = 0.5, if λ = λ 1 τ 2 (λ) = 0.4, if λ = λ 2 0, otherwise, 0, otherwise, τ1 (λ) = 0.7, if λ = λ 3 τ2 (λ) = 0.6, if λ = λ 4 0, otherwise, 0, otherwise. Then for 0 < r 0.4 the identity mapping f : (X, τ 1, τ 2 ) (X, τ 1, τ 2 ) is 21- fass closed but not 2-fuzzy closed, since if ν = 1 λ 2 we find that τ 2 (1 ν) = 0.4 τ 2 (1 f(ν)) = 0.0. Theorem 3.1. The surective mapping f : (X, τ 1, τ 2 ) (Y, τ1, τ2 ) from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ1, τ2 ) is -fagss closed if and only if for each µ I Y and λ r (τ, τ i ) F RO(X) such that f 1 (µ) λ, there exists ν r (τi, τ ) GF SSO(Y ) such that µ ν and f 1 (ν) λ. Proof. Necessity: Suppose that f is -fagss closed. Let µ I Y and λ r (τ, τ i ) F RO(X) such that f 1 (µ) λ. Since λ r (τ, τ i ) F RO(X) then 1 λ r (τ, τ i ) F RC(X) and since f is -fagss closed then f(1 λ) r (τi, τ ) GF SSC(Y ). Let ν = 1 f(1 λ). Then ν r (τ i, τ ) GF SSO(Y ). Since f 1 (µ) λ, 1 λ 1 f 1 (µ) = f 1 (1 µ). This implies that So, Also, we have f(1 λ) f(f 1 (1 µ)) 1 µ. µ 1 f(1 λ) = ν. f 1 (ν) = f 1 (1 f(1 λ)) = 1 f 1 (f(1 λ)) 1 (1 λ) = λ.

11 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 467 Sufficiency: Let η r (τ, τ i ) F RC(X) and µ = 1 f(η) I Y. Then f 1 (µ) = f 1 (1 f(η)) = 1 f 1 (f(η)) 1 η and since 1 η r (τ, τ i ) F RO(X), there exists ν r (τi, τ ) GF SSO(Y ) such that µ ν and f 1 (ν) 1 η. Since f is surective we have ν = f(f 1 (ν)) f(1 η) = 1 f(η). This implies that f(η) 1 ν. On the other hand since 1 f(η) = µ ν, then f(η) 1 ν. Thus f(η) = 1 ν r (τi, τ ) GF SSC(Y ). Hence f is -fagss closed mapping. Definition 3.2. Let f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) be a mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ 1, τ 2 ). Then f is called: (i) -fuzzy strongly semi open (briefly, -fss open) mapping if f(λ) r (τi, τ ) F SSO(Y ) for each λ IX with τ i (λ) r. (ii) -fuzzy almost open (briefly, -fa open) mapping if τi (f(λ)) r for each λ r (τ i, τ ) F RO(X). (iii) -fuzzy almost strongly semi-open (briefly, -fass open) mapping if f(λ) r (τi, τ ) F SSO(Y ) for each λ r (τ i, τ ) F RO(X). Remark 3.2. Both -fa openness and -fss openness imply -fass openness but the converse is not true in general as the following example shows: Example 3.4. Let X = {a, b, c, d} and Y = {x, y, z}. Define the fuzzy topologies τ 1, τ 2 : I X I and τ1, τ2 : I Y I as follows: 1, if λ = 0, 1 0.5, if λ {χ {c}, χ {d}, χ {c,d} } τ 1 (λ) = 0.7, if λ {χ {a,c}, χ {a,c,d} } 0, otherwise τ 2 (λ) = 0.5, if λ {χ {a}, χ {b}, χ {a,b} } 0, otherwise. 1, if µ = 0, 1 τ1 (µ) = 0.6, if µ = χ {z} τ2 (µ) = 0.4, if µ = χ {x,z} 0, otherwise, 0, otherwise. Then, for 0 < r 0.4 the mapping f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) which defined by f(a) = x, f(b) = f(d) = z, f(c) = y is 12-fass open but neither 12-fss open nor 12-fa open. Since, τ 1 (χ {c} ) = 0.5 > r, f(χ {c} ) = χ {y} r (τ 1, τ 2 ) F SSO(X) and χ {c,d} r (τ 1, τ 2 ) F RO(X), τ 1 (f(χ {c,d} )) = τ 1 (χ {y,z} ) = 0.0 r.

12 468 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF Definition 3.3. A mapping f : (X, τ 1, τ 2 ) (Y, τ1, τ2 ) from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ1, τ2 ) is said to be -fuzzy R-map if f 1 (µ) r (τ i, τ ) F RO(X) for every µ r (τi, τ ) F RO(Y ). Example 3.5. Let X = {a, b, c}. Define λ 1, λ 2, λ 3, λ 4 I X as follows: λ 1 (a) = 0.4 λ 1 (b) = 0.4 λ 1 (c) = 0.2 λ 2 (a) = 0.5 λ 2 (b) = 0.2 λ 2 (c) = 0.5 λ 3 (a) = 0.4 λ 3 (b) = 0.4 λ 3 (c) = 0.3 λ 4 (a) = 0.5 λ 4 (b) = 0.3 λ 4 (c) = 0.5 Define the fuzzy topologies τ 1, τ 2, τ1, τ2 : I X I as follows: τ 1 (λ) = 0.3, if λ = λ 1, λ 3 τ 2 (λ) = 0.3, if λ = λ 2 0, otherwise, 0, otherwise, τ1 (λ) = 0.4, if λ = λ 3 τ2 (λ) = 0.2, if λ = λ 4 0, otherwise, 0, otherwise. Then for 0 < r 0.3 the identity mapping f : (X, τ 1, τ 2 ) (X, τ 1, τ 2 ) is 12-fuzzy R-map. Theorem 3.2. Let f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) be a mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ 1, τ 2 ). Then f is -fuzzy R-map if it is fpc and -fass open. Proof. Let µ r (τi, τ ) F RO(Y ). Then τ i (µ) r. Since f is fpc, then τ i (f 1 (µ)) τi (µ) r and since f 1 (µ) C τ (f 1 (µ), r) then (1) f 1 (µ) = I τi (f 1 (µ), r) I τi (C τ (f 1 (µ), r), r). Since µ C τ (µ, r), we have f 1 (µ) f 1 (C τ (µ, r)). Since τ (1 f 1 (C τ (µ, r))) r, we have So, f 1 (µ) C τ (f 1 (µ), r) f 1 (C τ (µ, r)). f(i τi (C τ (f 1 (µ), r))) f(c τ (f 1 (µ), r)) f(f 1 (C τ (µ, r))) C τ (µ, r). Since f is -fass open and I τi (C τ (f 1 (µ), r), r) r (τ i, τ ) F RO(X), then f(i τi (C τ (f 1 (µ), r), r)) r (τi, τ ) F SSO(Y ). Thus f(i τi (C τ (f 1 (µ), r), r)) I τ s (C τ s (µ, r), r).

13 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 469 Since I τ s (C τ (µ, r), r) r (τ i, τ ) F SSO(Y ), then Then f(i τi (C τ (f 1 (µ), r), r)) I τ i (C τ (I τ i (C τ (µ, r), r), r), r) = I τ i (C τ (µ, r), r) = µ. (2) f 1 (µ) f 1 (f(i τi (C τ (f 1 (µ), r), r))) I τi (C τ (f 1 (µ), r), r). From (1) and (2) we have f 1 (µ) = I τi (C τ (f 1 (µ), r), r) this implies that f 1 (µ) r (τ i, τ ) F RO(X). Hence f is -fuzzy R-map. 4. Fuzzy binormal space In this section we introduce three concepts of fuzzy normality in fuzzy bitopological spaces namely, fuzzy binormality, fuzzy mildly binormality and fuzzy almost pairwise normality. Definition 4.1. A fbts (X, τ 1, τ 2 ) is said to be fuzzy binormal if η =C τi (η, r) qρ = C τ (ρ, r) implies there exist λ, µ I X with τ (λ) r, τ i (µ) r such that η λ, ρ µ and λ qµ. Theorem 4.1. Let (X, τ 1, τ 2 ) be a fbts. equivalent: Then the following statements are (i) (X, τ 1, τ 2 ) is fuzzy binormal. (ii) If η = C τi (η, r) qρ = C τ (ρ, r), then there exist λ r (τ i, τ ) GF SSO(X), µ r (τ, τ i ) GF SSO(X) such that η λ, ρ µ and λ qµ. (iii) For any η, µ I X with τ i (1 η) r, τ (µ) r and η µ there exist λ r (τ i, τ ) GF SSO(X) such that η λ (λ, r) µ. Proof. (i) (ii) It is easy. (ii) (iii) Let η, µ I X with τ i (1 η) r, τ (µ) r and η µ. Then τ i (1 η) r, τ (1 (1 µ)) r and η q(1 µ). By (ii) there exist λ r (τ i, τ ) GF SSO(X), ρ r (τ, τ i ) GF SSO(X) such that η λ, 1 µ ρ and λ qρ. Since ρ r (τ, τ i ) GF SSO(X), τ (µ) r and 1 µ ρ, then by Lemma 2.1, we have 1 µ I τ s (ρ, r). Since λ qρ, then λ 1 ρ 1 I τ s (ρ, r). So, Since λ 1 ρ, then So, η λ 1 ρ 1 I τ s (ρ, r). λ (λ, r) (1 ρ, r) = 1 I τ s (ρ, r). (from Theorem 2.3) η λ (λ, r) 1 I τ s (ρ, r) µ.

14 470 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF Thus η λ (λ, r) µ. (iii) (i) Let η = C τi (η, r) qρ = C τ (ρ, r). Then τ i (1 η) r, τ (1 ρ) r and η 1 ρ. By (iii) there exists λ r (τ i, τ ) GF SSO(X) such that This implies that η λ (λ, r) 1 ρ. ρ 1 (λ, r) = I τ s (1 λ, r). Since η λ, τ i (1 η) r and λ r (τ i, τ ) GF SSO(X), then by using Lemma 2.1, we have η I τ s (λ, r). Since I τ s (λ, r) r (τ, τ i ) F SSO(X) and I τ s (1 λ, r) r (τ i, τ ) F SSO(X), then and I τ s (λ, r) I τ (C τi (I τ (I τ s (λ, r), r), r), r) I τ s (1 λ, r) I τi (C τ (I τi (I τ s (1 λ, r), r), r), r). Put µ = I τ (C τi (I τ (I τ s (λ, r), r), r), r) and ν = I τi (C τ (I τi (I τ s (1 λ, r), r), r), r). Then τ (µ) r, τ i (ν) r, η µ, ρ ν and µ qν. Then (X, τ 1, τ 2 ) is fuzzy binormal. Theorem 4.2. Let f : (X, τ 1, τ 2 ) (Y, τ1, τ2 ) be fpc, fuzzy -fagss-closed and surection mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ1, τ2 ). If (X, τ 1, τ 2 ) is a fuzzy binormal space, then (Y, τ1, τ2 ) is also fuzzy binormal. Proof. Let η = C τ i (η, r) qρ = ρ τ (ρ, r), r I 0. Then τi (1 η) r, τ (1 ρ) r and η 1 ρ. Since f is fpc then, τ i (1 f 1 (η)) τi (1 η) r and τ (1 f 1 (ρ)) τ (1 ρ) r. Thus, f 1 (η) = C τi (f 1 (η), r) qf 1 (ρ) = C τ (f 1 (ρ), r). Since (X, τ 1, τ 2 ) is fuzzy binormal, there exist λ, µ I X with τ (λ) r, τ i (µ) r such that f 1 (η) λ, f 1 (ρ) µ and λ qµ. Let θ = I τ (C τi (λ, r), r) and δ = I τi (C τ (µ, r), r). Then θ r (τ, τ i ) F RO(X) and δ r (τ i, τ ) F RO(X). Also, f 1 (η) θ, f 1 (ρ) δ and θ qδ. By Theorem 3.1, there exist γ r (τi, τ ) GF SSO(Y ) and ν r (τ, τ i ) GF SSO(Y ) such that η γ, ρ ν, f 1 (γ) θ and f 1 (ν) δ. Since θ qδ, then γ qν and by Theorem 4.1, (Y, τ1, τ2 ) is fuzzy binormal. Definition 4.2. A fbts (X, τ 1, τ 2 ) is said to be fuzzy mildly binormal if for all η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) such that η qρ there exist λ, µ I X with τ (λ) r, τ i (µ) r such that η λ, ρ µ and λ qµ. Theorem 4.3. Let (X, τ 1, τ 2 ) be a fbts. equivalent: (i) (X, τ 1, τ 2 ) is fuzzy mildly binormal. Then the following statements are

15 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 471 (ii) For any η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) and η qρ, there exist λ r (τ i, τ ) GF SSO(X) and µ r (τ, τ i ) GF SSO(X) such that η λ, ρ µ and λ qµ. (iii) For any η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) and η qρ, there exist λ r (τ i, τ ) GF RSSO(X) and µ r (τ, τ i ) GF RSSO(X) such that η λ, ρ µ and λ qµ. (iv) For any η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RO(X) and η ρ, there exists λ r (τ i, τ ) GF RSSO(X) such that η λ (λ, r) ρ. (v) For any η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RO(X) and η ρ, there exists λ r (τ, τ i ) F SSO(X) such that η λ (λ, r) ρ. (vi) For any η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) and η qρ, there exist λ r (τ, τ i ) F SSO(X) and µ r (τ i, τ ) F SSO(X) such that η λ, ρ µ and λ qµ. Proof. (i) (ii) and (ii) (iii) are straightforward. (iii) (iv) Let η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RO(X) and η ρ. Consequently, η r (τ i, τ ) F RC(X), 1 ρ r (τ, τ i ) F RC(X) and η q(1 ρ). Then there exist λ r (τ i, τ ) GF RSSO(X) and µ r (τ, τ i ) GF RSSO(X) such that η λ, 1 ρ µ and λ qµ. By Lemma 2.1, we have 1 ρ I τ s (µ, r). Therefore we have η λ (λ, r) 1 I τ s (µ, r) ρ. Hence, η λ (λ, r) ρ. (iv) (v) Let η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RO(X) and η ρ. Then there exists µ r (τ i, τ ) GF RSSO(X) such that η µ (µ, r) ρ. Since η r (τ i, τ ) F RC(X) and η µ, then by using Lemma 2.1, we have η I τ s (µ, r). Put λ = I τ s (µ, r). Then λ r (τ, τ i ) F SSO(X) and η λ (λ, r) (µ, r) ρ. Hence, η λ (λ, r) ρ. (v) (vi) Let η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) and η qρ. Consequently, η r (τ i, τ ) F RC(X), 1 ρ r (τ, τ i ) F RO(X) and η 1 ρ. Then there exists λ r (τ, τ i ) F SSO(X) such that η λ (λ, r) 1 ρ. Then ρ 1 (λ, r). Put µ = 1 (λ, r). Then µ r (τ i, τ ) F SSO(X). Hence, η λ, ρ µ and λ qµ. (vi) (i) Let η r (τ i, τ ) F RC(X), ρ r (τ, τ i ) F RC(X) and η qρ. Then there exist λ r (τ, τ i ) F SSO(X) and µ r (τ i, τ ) F SSO(X) such that η λ, ρ µ and λ qµ. Put θ = I τ (C τi (I τ (λ, r), r), r) and δ = I τi (C τ (I τi (µ, r), r), r). Then we have, η θ, ρ δ and θ qδ. Hence, (X, τ 1, τ 2 ) is fuzzy mildly binormal.

16 472 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF Theorem 4.4. Let f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) be -fagss closed, -fuzzy R- map and surection mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ 1, τ 2 ). If (X, τ 1, τ 2 ) is a fuzzy mildly binormal space, then (Y, τ 1, τ 2 ) is also fuzzy mildly binormal. Proof. Let η r (τi, τ ) F RC(Y ), ρ r (τ i, τ ) F RC(Y ) and η qρ. Since f is -fuzzy R map then, f 1 (η) r (τ i, τ ) F RC(X), f 1 (ρ) r (τ i, τ ) F RC(X) furthermore, f 1 (η) qf 1 (ρ). Since (X, τ i, τ ) is fuzzy mildly binormal, there exist θ, δ I X with τ (θ) r, τ i (δ) r such that f 1 (η) θ,f 1 (ρ) δ and θ qδ. Put λ = I τ (C τi (θ, r)) and µ = I τi (C τ (δ, r)). Then λ r (τ, τ i ) F RO(X), µ r (τ i, τ ) F RO(X), f 1 (η) λ, f 1 (ρ) µ and λ qµ. By Theorem 3.1, there exist ξ r (τi, τ ) GF SSO(Y ), γ r (τ, τ i ) GF SSO(Y ) such that η ξ, f 1 (ξ) λ, ρ γ, f 1 (γ) µ. Also, we can find ξ qγ. Then, from Theorem 4.3, we have (Y, τ1, τ2 ) is also fuzzy mildly binormal. Definition 4.3. A fbts (X, τ 1, τ 2 ) is said to be fuzzy almost pairwise normal if for any η, ρ I X such that η r (τ i, τ ) F RC(X), τ (1 ρ) r and η qρ there exist λ, µ I X with τ (λ) r, τ i (µ) r such that η λ, ρ µ and λ qµ. Theorem 4.5. Let (X, τ 1, τ 2 ) be a fbts. Then the following statements are equivalent: (i) (X, τ 1, τ 2 ) is fuzzy almost pairwise normal. (ii) For any η r (τ i, τ ) F RC(X), ρ I X with τ (1 ρ) r and η qρ, there exist λ r (τ i, τ ) GF SSO(X) and µ r (τ, τ i ) GF SSO(X) such that η λ, ρ µ and λ qµ. (iii) For any η r (τ i, τ ) F RC(X), ρ I X with τ (1 ρ) r and η qρ, there exist λ r (τ i, τ ) GF RSSO(X) and µ r (τ, τ i ) GF RSSO(X) such that η λ, ρ µ and λ qµ. (iv) For any η r (τ i, τ ) F RC(X), ρ I X with τ (ρ) r and η ρ, there exists λ r (τ i, τ ) GF RSSO(X) such that η λ (λ, r) ρ. (v) For any η r (τ i, τ ) F RC(X), ρ I X with τ (ρ) r and η ρ, there exists λ r (τ, τ i ) F SSO(X) such that η λ (λ, r) ρ. (vi) For any η r (τ i, τ ) F RC(X), ρ I X with τ (1 ρ) r and η qρ, there exist λ r (τ, τ i ) F SSO(X) and µ r (τ i, τ ) F SSO(X) such that η λ, ρ µ and λ qµ. Proof. It is similar to the proof of Theorem 4.3. Theorem 4.6. Let f : (X, τ 1, τ 2 ) (Y, τ 1, τ 2 ) be -fass-open, -fass-closed, fpc and surection mapping from a fbts (X, τ 1, τ 2 ) to another fbts (Y, τ 1, τ 2 ). If (X, τ 1, τ 2 ) is a fuzzy almost pairwise normal, then (Y, τ 1, τ 2 ) is also fuzzy almost pairwise normal.

17 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 473 Proof. Let η r (τi, τ ) F RC(Y ), ρ IY with τ (ρ) r and η ρ. Since f is fpc and -fass open then, by Theorem 3.2, f 1 (η) r (τ i, τ ) F RC(X). Since f is fpc, then τ (f 1 (ρ)) τ (ρ) r. Furthermore, f 1 (η) f 1 (ρ). Since (X, τ 1, τ 2 ) is fuzzy almost pairwise normal and by Theorem 4.5(v), there exists λ r (τ, τ i ) F SSO(X) such that f 1 (η) λ (λ, r) f 1 (ρ). Since λ r (τ, τ i ) F SSO(X), then λ I τ (C τi (I τ (λ, r))) = µ (say). By using Theorem 2.4, we have Thus This implies that f 1 (η) λ µ C τi (µ, r) = C τi (I τ (λ, r), r)) = (λ, r) (f 1 (ρ), r) = f 1 (ρ). f 1 (η) µ C τi (µ, r) f 1 (ρ). f 1 (η) µ I τ (C τi (µ, r), r) C τi (µ, r) f 1 (ρ). Since f is surective we have η f(µ) f(i τ (C τi (µ, r), r)) f(c τi (µ, r)) f(f 1 (ρ)) = ρ. Since I τ (C τi (µ, r), r) r (τ, τ i ) F RO(X) and f is -fass-open, we have f(i τ (C τi (µ, r), r)) r (τ, τ i ) F SSO(Y ). Since C τ i (µ, r) = C τi (I τ (µ, r), r) r (τ i, τ ) F RC(X) and f is -fass-closed then, f(c τi (µ, r)) r (τ i, τ ) F SSC(Y ). Consequently Thus C τ s (f(c τ i (λ, r)), r) C τ s (f(c τ i (µ, r)), r) = f(c τi (µ, r)) ρ. η f(i τ (C τi (µ, r), r)) C τ s (f(i τ (C τi (µ, r), r), r) C τ s (f(c τ i (µ, r)), r) ρ. Then there exists f(i τ (C τi (µ, r), r)) r (τ, τ i ) F SSO(Y ) such that η f(i τ (C τi (µ, r), r)) C τ s (f(i τ (C τi (µ, r), r), r) ρ. Hence from Theorem 4.5(V), we have (Y, τ1, τ2 ) is fuzzy almost pairwise normal. Remark 4.1. From Definition 4.1, Definition 4.2 and Definition 4.3, we have the following implications: fuzzy mildly binormality fuzzy almost pairwise normality fuzzy binormality

18 474 A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF References [1] S. E. Abbas, A study of smooth topological spaces, Ph. D. Thesis, South Valley University, [2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), [3] K. C. Chattopadhyay, R. N. Hazra, and S. K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 (1992), no. 2, [4] K. C. Chattopadhyay and S. K. Samanta, Fuzzy topology: fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems 54 (1993), no. 2, [5] M. K. El Gayyar, E. E. Kerre, and A. A. Ramadan, Almost compactness and near compactness in smooth topological spaces, Fuzzy Sets and Systems 62 (1994), no. 2, [6] M. H. Ghanim, O. A. Tantawy, and F. M. Selim, Gradation of supra-openness, Fuzzy Sets and Systems 109 (2000), no. 2, [7] U. Höhle and A. P. Šostak, A general theory of fuzzy topological spaces, Fuzzy topology. Fuzzy Sets and Systems 73 (1995), no. 1, [8] Y. C. Kim, r-fuzzy semi-open sets in fuzzy bitopological spaces, Far East J. Math. Sci. (FJMS) Special Volume (2000), Part II, [9], δ-closure operators in fuzzy bitopological spaces, Far East J. Math. Sci. (FJMS) 2 (2000), no. 5, [10] Y. C. Kim, A. A. Ramadan, and S. E. Abbas, Some weakly forms of fuzzy continuous mappings in L-fuzzy bitopological spaces, J. Fuzzy Math. 9 (2001), no. 2, [11], Separation axioms in terms of θ-closure and δ-closure operators, Indian J. Pure Appl. Math. 34 (2003), no. 7, [12] T. Kubiak and A. P. Šostak, Lower set-valued fuzzy topologies, Quaestiones Math. 20 (1997), no. 3, [13] E. P. Lee, Preopen sets in smooth bitopological spaces, Commun. Korean Math. Soc. 18 (2003), no. 3, [14] P. M. Pu and Y. M. Liu, Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), no. 2, [15] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems 48 (1992), no. 3, [16] A. A. Ramadan and S. E. Abbas, On several types of continuity in fuzzy bitopological spaces, J. Fuzzy Math. 9 (2001), no. 2, [17] A. A. Ramadan, S. E. Abbas, and A. A. Abd El-latif, On fuzzy bitopological spaces in Šostak s sense, Commun. Korean Math. Soc. 21 (2006), no. 3, [18] A. P. Šostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo (2) Suppl. No. 11 (1985), [19], On the neighborhood structure of fuzzy topological spaces, Zb. Rad. No. 4 (1990), [20], Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), no. 6, Ahmed Abd El-Kader Ramadan Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef 62511, Egypt address: aramadan58@yahoo.com

19 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) 475 Salah El-Deen Abbas Mathematical Department Faculty of Science Sohag University Sohag 82524, Egypt address: sabbas73@yahoo.com Ahmed Aref Abd El-latif Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef 62511, Egypt address: ahmeda73@yahoo.com