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Gradation of supra-openness
Fuzzy Sets and Systems 109 (2000) 245–250
www.elsevier.com/locate/fss
Gradation of supra-openness M.H. Ghanim∗ , O.A. Tantawy, Fawzia M. Selim Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt Received November 1996; received in revised form October 1997
Abstract In 1992 a new deÿnition of a fuzzy topological space was given by Chattopadhyay et al. (1992) by introducing the concept of a gradation of openness. In this paper a new deÿnition of a fuzzy supra-topological (resp. supra-proximity, supra uniform) space is given by introducing the concept of a gradation of supra-openness (resp. supra-proximity, supra-uniformity) on a non-empty set X . How to construct a gradation of supra-openness induced by a gradation of supra-proximity (resp. supra-uniformity) is explained. The connection between gradations of supra-proximity and gradations of supra-uniformity are investigated. Throughout this paper, the class of all fuzzy subsets on a non-empty set X will be denoted by I X and fuzzy subsets will be denoted by small Greek letters. Also, we identify each subset with its characteristic function. ? 2000 Elsevier Science B.V. All rights reserved. AMS subject classiÿcation: 54 A 40 Keywords: Fuzzy topological spaces; Supra-fuzzy topological spaces; Fuzzy proximity spaces; Fuzzy uniform spaces
1. Fuzzy supra-topological spaces
clS and the fuzzy interior IntS are deÿned by
The concept of a (classical) fuzzy supra-topological space has been introduced as follows [3]. A collection S ⊂ I X is a fuzzy supra-topology on X if 0; 1 ∈ S and S is closed under arbitrary suprema. The pair (X; S) is a fuzzy supra-topological space. Members of S are the supra-open fuzzy sets and their complements are the supra-closed fuzzy sets.
clS =
Deÿnition 1.1. Let (X; S) be a (classical) fuzzy supratopological space. For ∈ I X , the fuzzy supra-closure
∗
Corresponding author.
^
Int S =
{ ∈ I X | is fuzzy
_
supra-closed set; ¿ }; { ∈ I X | is fuzzy supra-open; 6 }:
Deÿnition 1.2 (Ali El-Sheikh [3]). Let X 6= . An operator cl : I X → I X is said to be a (classical) fuzzy supra-closure operator on X , if it satisÿes the following axioms: (C1) cl(0) = 0, (C2) cl() ¿ , (C3) cl( ∨ ) ¿ cl() ∨ cl(), (C4) cl cl() = cl().
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 3 4 2 - 4
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M.H. Ghanim et al. / Fuzzy Sets and Systems 109 (2000) 245–250
Deÿnition 1.3 (Ali El-Sheikh [3]). An operator Int : I X → I X is said to be a (classical) fuzzy supra-interior operator on X (X 6= ) if it satisÿes the following axioms: (i) Int 0 = 0, (ii) Int 6 , (iii) Int ( ∧ ) 6 Int ∧ Int , (iv) Int(Int ) = Int . Let (X; S) be a (classical) fuzzy supra-topological space. For ∈ I X , if we call the value S() the grade of supra-openness of w.r.t. S, we note that the grades of supra-openness are either 0 or 1. In some sense, the fuzziness is absent in that classical deÿnition. To allow the grades of supra-openness to be of any value in the closed unit interval [0; 1], we introduce the concept of a gradation of supra-openness as follows: Deÿnition 1.4. Let X 6= . (a) A mapping S : I X → I which satisÿes (S1) S(0) V W = 1 = S(1), (S2) S( i i )¿ i S(i ) is called a gradation of supra-openness. The pair (X; S) is called a fuzzy supra-topological space. (b) A mapping C : I X → I which satisÿes (C1) C(0) V V = 1 = C(1), (C2) C( i i ) ¿ i C(i ) is called a gradation of supra-closedness. Proposition 1.1. Let S (resp. C) be a gradation of supra-openness (resp. closedness) on a non-empty set X and CS : I X → I (resp. SC : I X → I ) be a mapping deÿned by CS () = S(1−) (resp. SC () = C(1−)). Then CS (resp. SC ) is a gradation of supra-closedness (supra-openness) on X . Proof. It is a direct consequence of the above deÿnition. Corollary 1.1. Let S (resp. C) be a gradation of supra-openness (resp. closedness) on a non-empty set X . Then S = SCS (resp. C = CSC ). Proof. SCS () = CS (1 − ) = S(1 − (1 − )) = S() ∀ ∈ I X i.e., SCS = S. Similarly, we prove that CSC = C.
Remark 1.1. If all grades of supra-openness are restricted to be 0 or 1, then Deÿnition 1.4 is reduced to the deÿnition of a (classical) fuzzy supra-topological space. In such a case, we say that the new concept is a good extension for the (classical) fuzzy case. In this paper, we refer to fuzzy supra-topology in the sense of [3] by a (classical) fuzzy supra-topology. Also, by a (classical) supra-topology we refer to the well-known (crisp) supra-topology. Proposition 1.2. Let (X; S) be a fuzzy supratopological space. Then; for each r ∈ (0; 1]; the family Sr = { ∈ I X | S() ¿ r} is a (classical ) fuzzy supra-topology on X . Proof. Straightforward. Deÿnition 1.5. For each r ∈ (0; 1], the family Sr is called the r-level (classical) fuzzy supra-topology on X, of the gradation of supra-openness S. Proposition 1.3. Let (X; S) be a fuzzy supratopological space. Then; the family {Sr | r ∈ (0; 1]} of all r-level (classical ) fuzzy supra-topologies T on X; of S is a descending family. Furthermore; Sr = t¡r St . Proof. That the family {Sr | r ∈ (0; 1]} is a descending family is straightforward. We prove the second asConsequently, sertion. T For r¿t, it is clear that Sr ⊂ St . T Sr ⊂ r¿t St . Suppose ∃ ∈= Sr s.t. ∈ r¿t St . Since S()¡r, then, ∃t ∈ (0; 1] s.t. S()¡t¡r. In T other . Consequently, ∈ 6 words, ∃t¡r s.t. ∈ = S t r¿t St . T Hence Sr = r¿t St . 2. Fuzzy supra-proximity spaces The concept of a (classical) fuzzy supra-proximity space has been deÿned [3] as follows: A binary relation on I X is called a (classical) fuzzy supra-proximity on X i it satisÿes the following axioms: (SP1) (; ) ∈ ⇒ (; ) ∈ , (SP2) (; ) ∈ or (; ) ∈ ⇒ (; ∨ ) ∈ , (SP3) (0; 1) 6∈ , (SP4) (; ) 6∈ ⇒ ∃ ∈ I X s.t. (; ) 6∈ , ((1 − ); ) 6∈ , (SP5) (; ) ∈= ⇒ 6 1 − .
M.H. Ghanim et al. / Fuzzy Sets and Systems 109 (2000) 245–250
If is a (classical) fuzzy supra-proximity on X , then the pair (X; ) is called a (classical) fuzzy supra-proximity space. Each (classical) fuzzy supraproximity on X induces a (classical) fuzzy supratopology on X as follows: Let (X; ) be a (classical) fuzzy supra-proximity. Let cl : I X → I X be deÿned V X by cl () = { ∈ I | (; 1 − ) 6∈ }. Then, cl is a (classical) fuzzy supra-closure operator. Consequently, the family = { ∈ I X | cl (1 − ) = 1 − } is a (classical) fuzzy supra-topology on X . For (; ) ∈ I X × I X if we call the value (; ) the grade of supra-proximity of (; ) w.r.t. , we notice that the grades of supra-proximity are either 0 or 1. In some sense, the fuzziness is absent in the above deÿnition. To allow grades of supra-proximity to be any value in the closed unit interval I = [0; 1], we introduce the concept of a gradation of supra-proximity as follows. Deÿnition 2.1. A mapping from I X × I X into I is said to be a gradation of supra-proximity on X, if it satisÿes the following axioms: (SFP1) (; ) = (; ), (SFP2) (; ) ∨ (; ) 6 (; ∨ ), (SFP3) (0; 1) = 0, (SFP4) (; )¡ r ⇒ ∃ ∈ I X s.t. (; )¡r, (1 − ; )¡r, (SFP5) (; ) 6= 1 ⇒ 6 1 − . The pair (X; ) is called a fuzzy supra-proximity space. Remark 2.1. If all grades of supra-proximity are restricted to be 0 or 1, then Deÿnition 2.1 is reduced to the deÿnition of a (classical) fuzzy supra-proximity space. Hence, the new concept is a good extension for the (classical) fuzzy case. We refer to fuzzy supra-proximity in the sense of [3] by a (classical) fuzzy supra-proximity. Also, we refer to the well-known (crisp) supra-proximity by a (classical) supra-proximity. Proposition 2.1. Let (X; ) be a fuzzy supraproximity space. Then; for each r ∈ (0; 1]; the family r = {(; ) | (; ) ¿ r} is a (classical) fuzzy supra-proximity on X . Proof. Given r ∈ (0; 1].
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(SP1) Let (; ) ∈ r . Then, (; ) ¿ r. From (SFP1), we have (; ) ¿ r. Thus, (; ) ∈ r . (SP2) Let (; ) ∈ r or (; ) ∈ r . Then, (; ) ¿ r or (; ) ¿ r. Hence, (; ) ∨ (; ) ¿ r. From (SFP2), we have (; ∨ ) ¿ r. Consequently, (; ∨ ) ∈ r . (SP3) Since (0; 1) = 0, then (0; 1)¡r. Hence, (0; 1) ∈= r . (SP4) Let (; ) 6∈ r . Then, (; )¡r. From (SFP4), ∃ ∈I X s.t. (; )¡r; (1 − ; )¡r. Consequently, ∃ ∈I X s.t. (; )6∈ r , ((1−); )6∈ r . Hence, r is a (classical) fuzzy supra-proximity on X ∀r ∈(0; 1]. Deÿnition 2.2. For each r ∈(0; 1]; r is called the rlevel (classical) fuzzy supra-proximity on X , of the gradation of supra-proximity . Proposition 2.2. Let (X; ) be a fuzzy supraproximity space. The family {r | r ∈(0; 1]} of all r-level (classical) fuzzy supra-proximities T of ; is a descending family. Furthermore, r = t¡r t . Proof. Let t¡r and (; )∈r . Then (; T ) ¿ r¿t. Hence (; ) ∈T . Consequently, ⊆ t r t¡r t . Let (; )∈ t¡r t and assume that (; ) 6∈ r . Then (; )¡r. T )¡t. T Consequently, ∃t¡r; (; Thus, (; )6∈ t¡r t . Hence, we have r = t¡r t . In what follows, we explain how a gradation of fuzzy supra-proximity induces a gradation of supra openness. Deÿnition 2.3. Let (X; ) be a fuzzy supra-proximity space. For each r ∈(0; 1], let Intr : I X → I X be deÿned as follows: For ∈I X ; Intr = sup{∈I X | (; 1 − )¡r}: The family {Int r | r ∈(0; 1]} is called the fuzzy interior operators of the fuzzy supra-proximity space (X; ). Proposition 2.3. The interior fuzzy operators of the fuzzy supra-proximity space (X; ) have the following properties: (i) Int r 1 = 1 ∀r ∈(0; 1]; (ii) Int r 6 ;
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(iii) Intr Int r = Intr ; (iv) Int r ( ∧ ) 6 Intr ∧ Intr ; (v) r 6 r 0 ⇒ Intr 6 Intr 0 ; where ; ∈I X ; r; r 0 ∈ (0; 1]. W Proof. (i) Int r 1 = { ∈I X | (; 0)¡r} = sup I X =1. W (ii) Intr = { ∈I X | (; 1 − )¡r}. For any ∈I X , if (; 1 − )¡r, W then (; 1 − ) 6= 1. Consequently, 6 . Thus, { ∈I X | (; 1 − ) ¡r} 6 . Consequently, Int r 6 . (iii) From (ii), we have Intr (Intr ) 6 Intr . Conversely, let ∈I X s.t. (; 1−)¡r (i.e. 6 Int r ). Then ∃ ∈I X s.t. (; )¡r; (1 − ; 1 − )¡r. Hence 6 Intr (1−) and (1 −) 6 Int r . Consequently, 6 Intr (1−) 6 Intr Intr . In other words, Int r () 6 Intr Int r . Consequently Int r Int r = Intr . (iv) Obvious. 0 (v) Let r 6 r 0 and ∈I X s.t. W (; 1X− )¡r 6 r . {∈I | (; 1 − ) Then, W6 Int r 0 (). Hence, ¡r} 6 {∈I X | (; 1 − )¡r 0 }. Consequently, Int r 6 Intr 0 . Theorem 2.1. Let (X; ) be a fuzzy supra-proximity space. The mapping S : I X → I deÿned by S () = W {r | Int r = } is a gradation of supra-openness on X . It is called the gradation of supra-openness induced by the gradation of supra-proximity . Proof. It is a direct consequence of Proposition 2.3 and Deÿnition 1.1.
3. Fuzzy supra-uniform spaces The concept of a (classical) supra-uniformity on a non-empty set X has been deÿned as follows (cf. [3]) A (classical) supra-uniformity on a set X 6= is a non-empty collection U of subsets of X ×X satisfying the following conditions: (U1) X ⊆ u ∀u ∈U , (U2) u ∈U; u ⊆ ⇒ ∈U , (U3) u ∈U ⇒ ∃∈U s.t. ◦ ⊆ u, (U4) u ∈U ⇒ u−1 ∈U . The pair (X; U ) is called a (classical) supra-uniform space. In [2] it was shown that, there is a 1–1 corre-
spondence between sets containing the diagonal and mappings satisfying certain conditions, in the following way: Let u ⊂ X ×X which contains the diagonal X . Deÿne a map u : P(X ) → P(X ) by u(A) = {y | (x; y) ∈u for some x ∈A}. ItSis obvious S that (1) u( ) = , (2) u(A) ⊇ A (3) u( A ) = (u(A )). Conversely, let u : P(X ) → P(X ) satisÿes the above conditions (1) – (3). Deÿne u ⊆ X × X by u = {(x; y) | y ∈u({x})}. It is clear that u contains the diagonal X . Identifying each set u containing the diagonal X , with its corresponding mapping u : P(X ) → P(X ), which satisÿes the conditions (1) – (3), the (classical) supra-uniform space was re-deÿned as follows: Let D be the set of maps u : P(X ) → P(X ) which satisÿes the above conditions (1) – (3). A (classical) supra-uniformity on X is a subset U of D such that (SU1) U 6= , (SU2) u ∈U ⇒ u−1 ∈U , (SU3) u ∈U; u ⊆ ⇒ ∈U , (SU4) u ∈U ⇒ ∃ ∈U s.t. ◦ ⊆ u. In [2] the concept of supra-uniformity has been generalized to fuzzy case as follows: let D be the I X whichWsatisÿes (1) u(0) = 0, set of maps u : I X →W (2) u() ¿ ; (3) u( i i ) = i (u(i )). A (classical) fuzzy supra-uniformity on X is a subset U of D which satisÿes (FSU1) U 6= , (FSU2) u ∈ U ⇒ u−1 ∈ U; (FSU3) u ∈ U; u 6 ⇒ ∈ U; (FSU4) u ∈ U ⇒ ∃ ∈ U s.t. ◦ 6 u. where u−1 : I X → I X deÿned by u−1 () = Inf { ∈ I X | u(1 − ) 6 1 − }. A (classical) fuzzy supra-unformity U can be written as a function from D into {0; 1} which satisÿes (FSU1)0 ∃u ∈D s.t. U (u)¿0, (FSU2)0 U (u)¿0 ⇒ U (u−1 )¿0, (FSU3)0 U (u)¿0 ⇒ ∃ ∈D s.t. U ()¿0, ◦ 6 u, (FSU4)0 U (u)¿0; u 6 ⇒ U ()¿0. For any u ∈D, if we call the value U (u) the grade of supra-uniformity of u w.r.t. the (classical) fuzzy supra-uniformity U , we notice that the grades of suprauniformity of any u ∈D is either 0 or 1. In some sense, the fuzziness is absent. To allow the grades of fuzzy supra-uniformity to be any value in I = [0; 1], we introduce the concept of a gradation of supra-uniformity as follows:
M.H. Ghanim et al. / Fuzzy Sets and Systems 109 (2000) 245–250
Deÿnition 3.1. Let D be the set of all maps u : I X → I X which satisÿes the above conditions (1)0 – (3)0 . A mapping U from D into I is said to be a gradation of supra-uniformity if it satisÿes the following axioms: (GSU1) ∃u ∈D s.t. U (u) = 1, (GSU2) U (u) = U (u−1 ); (GSU3) U (u) ¿ r; u 6 ⇒ U () ¿ r, and (GSU4) U (u) ¿ r ⇒ ∃∈D s.t. U () ¿ r; ◦ 6 u, = (()) where ◦ : I X → I X deÿned by ( ◦ )()V and u−1 : I X → I X deÿned by u−1 () = { ∈I X | u(1 − ) 6 1 − }. The pair (X; U ) is called a fuzzy supra-uniform space. Remark 3.1. If all grades of supra-uniformity are restricted to be 0 or 1, then Deÿnition 3.1 is reduced to the deÿnition of a (classical) fuzzy supra-uniform space. In other words, the new concept is a good extension for the (classical) fuzzy case. Proposition 3.1. Let (X; U ) be a fuzzy suprauniform space. Then, ∀r ∈(0; 1]; the family Ur = {u ∈D | U (u) ¿ r} is a (classical) fuzzy supra-uniformity on X . Proof. (SU1) From (GSU1) ∃u ∈D s.t. U (u) = 1. In other words ∃u ∈D s.t. U (u) ¿ r ∀r. Hence Ur 6= . (SU2) Let u ∈Ur . Then U (u−1 ) = U (u) ¿ r. Hence, u−1 ∈Ur . (SU3) Let u ∈Ur , u 6 . From (GSU3), we have U (u) ¿ r; u 6 . Thus, U () ¿ r. Consequently, ∈Ur . (SU4) Let u ∈Ur . Then, U (u) ¿ r. From (GSU 4), ∃ ∈D s.t. U () ¿ r; ◦ 6 u. Thus, ∃ ∈Ur ; ◦ 6 u. Hence, we have the result. Deÿnition 3.2. For each r ∈(0; 1], the family Ur is called the r-level (classical) fuzzy supra-uniformity of the gradation of supra-uniformity U . Proposition 3.2. Let (X; U ) be a fuzzy suprauniform space. The family {Ur : r ∈(0; 1]} of all r-level (classical) fuzzy supra-uniformities on X; w.r.t. U is a descending family. Furthermore, Ur = T U t. t¡r Proof. Let r ∈(0; 1]; t¡r andTu ∈Ur . Then, U (u) ¿ r¿t and u ∈Ut . Thus, Ur ⊂ t¡r Ut . Suppose that
249
T there exists u ∈ t¡r Ut s.t. u 6∈ Ur . Then, U (u)¡r. This impliesT that ∃t¡r s.t. U (u)¡t¡r. Consequently, u 6∈ t¡r Ut ; this is a contradiction. Hence, we have the result. Proposition 3.3. Let (X; U ) be a (classical) fuzzy supra-uniform space. Then, the family U ; deÿned by i _ = { ∈I X | u() 6 for some u ∈U };
∈ U
is a (classical) fuzzy supra-topology on X . Proof. (1) Since u(1) ¿ 1 ∀u ∈U , then sup{ ∈I X |u() 6 1 for some u ∈U } = 1; i.e. 1 ∈ U . Since u(0) = 0 and u() ¿ ∀ ∈I X ; then ∈U } = 0, i.e. 0∈ U . sup{ ∈I X |u() 6 0 for some uW (2) Let ∈ U ∀ . Then, = {∈I X | u() 6 for some u ∈U }. Thus, __ _ = ∈I X | u() 6 for some u ∈U
= Thus,
_
W
) _ ∈ I u() 6 for some u ∈ U :
(
X
∈ U .
Deÿnition 3.3. Let (X; U ) be a (classical) fuzzy supra-uniform space. The family U , as deÿned above, is a (classical) fuzzy supra-topology induced by U . The pair (X; U ) is the (classical) fuzzy supratopological space induced by (X; U ). Proposition 3.4. Let (X; U ) be a fuzzy suprauniform space. For r ∈(0; 1]; let Ur be the (classical) fuzzy supra-topology induced by Ur . Then the familyT{ Ur | r ∈(0; 1]} is descending. Furthermore, Ur = s¡r Us . Proof. It is similar to that given in [1, Proposition 2.11] Proposition 3.5. Let (X; U ) be a fuzzy supraX uniform space. W The mapping U : I → I deÿned by U () = {r ∈(0; 1] | ∈ Ur } is a gradation of supra-openness on X . It is called the gradation of
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M.H. Ghanim et al. / Fuzzy Sets and Systems 109 (2000) 245–250
supra-openness induced by the gradation of suprauniformity U . Proof. It is similar to that given for [1, Proposition 2.12]. References [1] K.C. Chattopadhyay, R.H. Hazra, S.K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 (1992) 236 –292.
[2] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. (58) 559 –571. [3] S.A. Aly El-Sheikh, Some strong properties of fuzzy bitopological spaces, Ph.D. Thesis, Ain Shams University, Faculty of Education, Egypt, 1995.
www.elsevier.com/locate/fss
Gradation of supra-openness M.H. Ghanim∗ , O.A. Tantawy, Fawzia M. Selim Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt Received November 1996; received in revised form October 1997
Abstract In 1992 a new deÿnition of a fuzzy topological space was given by Chattopadhyay et al. (1992) by introducing the concept of a gradation of openness. In this paper a new deÿnition of a fuzzy supra-topological (resp. supra-proximity, supra uniform) space is given by introducing the concept of a gradation of supra-openness (resp. supra-proximity, supra-uniformity) on a non-empty set X . How to construct a gradation of supra-openness induced by a gradation of supra-proximity (resp. supra-uniformity) is explained. The connection between gradations of supra-proximity and gradations of supra-uniformity are investigated. Throughout this paper, the class of all fuzzy subsets on a non-empty set X will be denoted by I X and fuzzy subsets will be denoted by small Greek letters. Also, we identify each subset with its characteristic function. ? 2000 Elsevier Science B.V. All rights reserved. AMS subject classiÿcation: 54 A 40 Keywords: Fuzzy topological spaces; Supra-fuzzy topological spaces; Fuzzy proximity spaces; Fuzzy uniform spaces
1. Fuzzy supra-topological spaces
clS and the fuzzy interior IntS are deÿned by
The concept of a (classical) fuzzy supra-topological space has been introduced as follows [3]. A collection S ⊂ I X is a fuzzy supra-topology on X if 0; 1 ∈ S and S is closed under arbitrary suprema. The pair (X; S) is a fuzzy supra-topological space. Members of S are the supra-open fuzzy sets and their complements are the supra-closed fuzzy sets.
clS =
Deÿnition 1.1. Let (X; S) be a (classical) fuzzy supratopological space. For ∈ I X , the fuzzy supra-closure
∗
Corresponding author.
^
Int S =
{ ∈ I X | is fuzzy
_
supra-closed set; ¿ }; { ∈ I X | is fuzzy supra-open; 6 }:
Deÿnition 1.2 (Ali El-Sheikh [3]). Let X 6= . An operator cl : I X → I X is said to be a (classical) fuzzy supra-closure operator on X , if it satisÿes the following axioms: (C1) cl(0) = 0, (C2) cl() ¿ , (C3) cl( ∨ ) ¿ cl() ∨ cl(), (C4) cl cl() = cl().
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 3 4 2 - 4
246
M.H. Ghanim et al. / Fuzzy Sets and Systems 109 (2000) 245–250
Deÿnition 1.3 (Ali El-Sheikh [3]). An operator Int : I X → I X is said to be a (classical) fuzzy supra-interior operator on X (X 6= ) if it satisÿes the following axioms: (i) Int 0 = 0, (ii) Int 6 , (iii) Int ( ∧ ) 6 Int ∧ Int , (iv) Int(Int ) = Int . Let (X; S) be a (classical) fuzzy supra-topological space. For ∈ I X , if we call the value S() the grade of supra-openness of w.r.t. S, we note that the grades of supra-openness are either 0 or 1. In some sense, the fuzziness is absent in that classical deÿnition. To allow the grades of supra-openness to be of any value in the closed unit interval [0; 1], we introduce the concept of a gradation of supra-openness as follows: Deÿnition 1.4. Let X 6= . (a) A mapping S : I X → I which satisÿes (S1) S(0) V W = 1 = S(1), (S2) S( i i )¿ i S(i ) is called a gradation of supra-openness. The pair (X; S) is called a fuzzy supra-topological space. (b) A mapping C : I X → I which satisÿes (C1) C(0) V V = 1 = C(1), (C2) C( i i ) ¿ i C(i ) is called a gradation of supra-closedness. Proposition 1.1. Let S (resp. C) be a gradation of supra-openness (resp. closedness) on a non-empty set X and CS : I X → I (resp. SC : I X → I ) be a mapping deÿned by CS () = S(1−) (resp. SC () = C(1−)). Then CS (resp. SC ) is a gradation of supra-closedness (supra-openness) on X . Proof. It is a direct consequence of the above deÿnition. Corollary 1.1. Let S (resp. C) be a gradation of supra-openness (resp. closedness) on a non-empty set X . Then S = SCS (resp. C = CSC ). Proof. SCS () = CS (1 − ) = S(1 − (1 − )) = S() ∀ ∈ I X i.e., SCS = S. Similarly, we prove that CSC = C.
Remark 1.1. If all grades of supra-openness are restricted to be 0 or 1, then Deÿnition 1.4 is reduced to the deÿnition of a (classical) fuzzy supra-topological space. In such a case, we say that the new concept is a good extension for the (classical) fuzzy case. In this paper, we refer to fuzzy supra-topology in the sense of [3] by a (classical) fuzzy supra-topology. Also, by a (classical) supra-topology we refer to the well-known (crisp) supra-topology. Proposition 1.2. Let (X; S) be a fuzzy supratopological space. Then; for each r ∈ (0; 1]; the family Sr = { ∈ I X | S() ¿ r} is a (classical ) fuzzy supra-topology on X . Proof. Straightforward. Deÿnition 1.5. For each r ∈ (0; 1], the family Sr is called the r-level (classical) fuzzy supra-topology on X, of the gradation of supra-openness S. Proposition 1.3. Let (X; S) be a fuzzy supratopological space. Then; the family {Sr | r ∈ (0; 1]} of all r-level (classical ) fuzzy supra-topologies T on X; of S is a descending family. Furthermore; Sr = t¡r St . Proof. That the family {Sr | r ∈ (0; 1]} is a descending family is straightforward. We prove the second asConsequently, sertion. T For r¿t, it is clear that Sr ⊂ St . T Sr ⊂ r¿t St . Suppose ∃ ∈= Sr s.t. ∈ r¿t St . Since S()¡r, then, ∃t ∈ (0; 1] s.t. S()¡t¡r. In T other . Consequently, ∈ 6 words, ∃t¡r s.t. ∈ = S t r¿t St . T Hence Sr = r¿t St . 2. Fuzzy supra-proximity spaces The concept of a (classical) fuzzy supra-proximity space has been deÿned [3] as follows: A binary relation on I X is called a (classical) fuzzy supra-proximity on X i it satisÿes the following axioms: (SP1) (; ) ∈ ⇒ (; ) ∈ , (SP2) (; ) ∈ or (; ) ∈ ⇒ (; ∨ ) ∈ , (SP3) (0; 1) 6∈ , (SP4) (; ) 6∈ ⇒ ∃ ∈ I X s.t. (; ) 6∈ , ((1 − ); ) 6∈ , (SP5) (; ) ∈= ⇒ 6 1 − .
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If is a (classical) fuzzy supra-proximity on X , then the pair (X; ) is called a (classical) fuzzy supra-proximity space. Each (classical) fuzzy supraproximity on X induces a (classical) fuzzy supratopology on X as follows: Let (X; ) be a (classical) fuzzy supra-proximity. Let cl : I X → I X be deÿned V X by cl () = { ∈ I | (; 1 − ) 6∈ }. Then, cl is a (classical) fuzzy supra-closure operator. Consequently, the family = { ∈ I X | cl (1 − ) = 1 − } is a (classical) fuzzy supra-topology on X . For (; ) ∈ I X × I X if we call the value (; ) the grade of supra-proximity of (; ) w.r.t. , we notice that the grades of supra-proximity are either 0 or 1. In some sense, the fuzziness is absent in the above deÿnition. To allow grades of supra-proximity to be any value in the closed unit interval I = [0; 1], we introduce the concept of a gradation of supra-proximity as follows. Deÿnition 2.1. A mapping from I X × I X into I is said to be a gradation of supra-proximity on X, if it satisÿes the following axioms: (SFP1) (; ) = (; ), (SFP2) (; ) ∨ (; ) 6 (; ∨ ), (SFP3) (0; 1) = 0, (SFP4) (; )¡ r ⇒ ∃ ∈ I X s.t. (; )¡r, (1 − ; )¡r, (SFP5) (; ) 6= 1 ⇒ 6 1 − . The pair (X; ) is called a fuzzy supra-proximity space. Remark 2.1. If all grades of supra-proximity are restricted to be 0 or 1, then Deÿnition 2.1 is reduced to the deÿnition of a (classical) fuzzy supra-proximity space. Hence, the new concept is a good extension for the (classical) fuzzy case. We refer to fuzzy supra-proximity in the sense of [3] by a (classical) fuzzy supra-proximity. Also, we refer to the well-known (crisp) supra-proximity by a (classical) supra-proximity. Proposition 2.1. Let (X; ) be a fuzzy supraproximity space. Then; for each r ∈ (0; 1]; the family r = {(; ) | (; ) ¿ r} is a (classical) fuzzy supra-proximity on X . Proof. Given r ∈ (0; 1].
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(SP1) Let (; ) ∈ r . Then, (; ) ¿ r. From (SFP1), we have (; ) ¿ r. Thus, (; ) ∈ r . (SP2) Let (; ) ∈ r or (; ) ∈ r . Then, (; ) ¿ r or (; ) ¿ r. Hence, (; ) ∨ (; ) ¿ r. From (SFP2), we have (; ∨ ) ¿ r. Consequently, (; ∨ ) ∈ r . (SP3) Since (0; 1) = 0, then (0; 1)¡r. Hence, (0; 1) ∈= r . (SP4) Let (; ) 6∈ r . Then, (; )¡r. From (SFP4), ∃ ∈I X s.t. (; )¡r; (1 − ; )¡r. Consequently, ∃ ∈I X s.t. (; )6∈ r , ((1−); )6∈ r . Hence, r is a (classical) fuzzy supra-proximity on X ∀r ∈(0; 1]. Deÿnition 2.2. For each r ∈(0; 1]; r is called the rlevel (classical) fuzzy supra-proximity on X , of the gradation of supra-proximity . Proposition 2.2. Let (X; ) be a fuzzy supraproximity space. The family {r | r ∈(0; 1]} of all r-level (classical) fuzzy supra-proximities T of ; is a descending family. Furthermore, r = t¡r t . Proof. Let t¡r and (; )∈r . Then (; T ) ¿ r¿t. Hence (; ) ∈T . Consequently, ⊆ t r t¡r t . Let (; )∈ t¡r t and assume that (; ) 6∈ r . Then (; )¡r. T )¡t. T Consequently, ∃t¡r; (; Thus, (; )6∈ t¡r t . Hence, we have r = t¡r t . In what follows, we explain how a gradation of fuzzy supra-proximity induces a gradation of supra openness. Deÿnition 2.3. Let (X; ) be a fuzzy supra-proximity space. For each r ∈(0; 1], let Intr : I X → I X be deÿned as follows: For ∈I X ; Intr = sup{∈I X | (; 1 − )¡r}: The family {Int r | r ∈(0; 1]} is called the fuzzy interior operators of the fuzzy supra-proximity space (X; ). Proposition 2.3. The interior fuzzy operators of the fuzzy supra-proximity space (X; ) have the following properties: (i) Int r 1 = 1 ∀r ∈(0; 1]; (ii) Int r 6 ;
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(iii) Intr Int r = Intr ; (iv) Int r ( ∧ ) 6 Intr ∧ Intr ; (v) r 6 r 0 ⇒ Intr 6 Intr 0 ; where ; ∈I X ; r; r 0 ∈ (0; 1]. W Proof. (i) Int r 1 = { ∈I X | (; 0)¡r} = sup I X =1. W (ii) Intr = { ∈I X | (; 1 − )¡r}. For any ∈I X , if (; 1 − )¡r, W then (; 1 − ) 6= 1. Consequently, 6 . Thus, { ∈I X | (; 1 − ) ¡r} 6 . Consequently, Int r 6 . (iii) From (ii), we have Intr (Intr ) 6 Intr . Conversely, let ∈I X s.t. (; 1−)¡r (i.e. 6 Int r ). Then ∃ ∈I X s.t. (; )¡r; (1 − ; 1 − )¡r. Hence 6 Intr (1−) and (1 −) 6 Int r . Consequently, 6 Intr (1−) 6 Intr Intr . In other words, Int r () 6 Intr Int r . Consequently Int r Int r = Intr . (iv) Obvious. 0 (v) Let r 6 r 0 and ∈I X s.t. W (; 1X− )¡r 6 r . {∈I | (; 1 − ) Then, W6 Int r 0 (). Hence, ¡r} 6 {∈I X | (; 1 − )¡r 0 }. Consequently, Int r 6 Intr 0 . Theorem 2.1. Let (X; ) be a fuzzy supra-proximity space. The mapping S : I X → I deÿned by S () = W {r | Int r = } is a gradation of supra-openness on X . It is called the gradation of supra-openness induced by the gradation of supra-proximity . Proof. It is a direct consequence of Proposition 2.3 and Deÿnition 1.1.
3. Fuzzy supra-uniform spaces The concept of a (classical) supra-uniformity on a non-empty set X has been deÿned as follows (cf. [3]) A (classical) supra-uniformity on a set X 6= is a non-empty collection U of subsets of X ×X satisfying the following conditions: (U1) X ⊆ u ∀u ∈U , (U2) u ∈U; u ⊆ ⇒ ∈U , (U3) u ∈U ⇒ ∃∈U s.t. ◦ ⊆ u, (U4) u ∈U ⇒ u−1 ∈U . The pair (X; U ) is called a (classical) supra-uniform space. In [2] it was shown that, there is a 1–1 corre-
spondence between sets containing the diagonal and mappings satisfying certain conditions, in the following way: Let u ⊂ X ×X which contains the diagonal X . Deÿne a map u : P(X ) → P(X ) by u(A) = {y | (x; y) ∈u for some x ∈A}. ItSis obvious S that (1) u( ) = , (2) u(A) ⊇ A (3) u( A ) = (u(A )). Conversely, let u : P(X ) → P(X ) satisÿes the above conditions (1) – (3). Deÿne u ⊆ X × X by u = {(x; y) | y ∈u({x})}. It is clear that u contains the diagonal X . Identifying each set u containing the diagonal X , with its corresponding mapping u : P(X ) → P(X ), which satisÿes the conditions (1) – (3), the (classical) supra-uniform space was re-deÿned as follows: Let D be the set of maps u : P(X ) → P(X ) which satisÿes the above conditions (1) – (3). A (classical) supra-uniformity on X is a subset U of D such that (SU1) U 6= , (SU2) u ∈U ⇒ u−1 ∈U , (SU3) u ∈U; u ⊆ ⇒ ∈U , (SU4) u ∈U ⇒ ∃ ∈U s.t. ◦ ⊆ u. In [2] the concept of supra-uniformity has been generalized to fuzzy case as follows: let D be the I X whichWsatisÿes (1) u(0) = 0, set of maps u : I X →W (2) u() ¿ ; (3) u( i i ) = i (u(i )). A (classical) fuzzy supra-uniformity on X is a subset U of D which satisÿes (FSU1) U 6= , (FSU2) u ∈ U ⇒ u−1 ∈ U; (FSU3) u ∈ U; u 6 ⇒ ∈ U; (FSU4) u ∈ U ⇒ ∃ ∈ U s.t. ◦ 6 u. where u−1 : I X → I X deÿned by u−1 () = Inf { ∈ I X | u(1 − ) 6 1 − }. A (classical) fuzzy supra-unformity U can be written as a function from D into {0; 1} which satisÿes (FSU1)0 ∃u ∈D s.t. U (u)¿0, (FSU2)0 U (u)¿0 ⇒ U (u−1 )¿0, (FSU3)0 U (u)¿0 ⇒ ∃ ∈D s.t. U ()¿0, ◦ 6 u, (FSU4)0 U (u)¿0; u 6 ⇒ U ()¿0. For any u ∈D, if we call the value U (u) the grade of supra-uniformity of u w.r.t. the (classical) fuzzy supra-uniformity U , we notice that the grades of suprauniformity of any u ∈D is either 0 or 1. In some sense, the fuzziness is absent. To allow the grades of fuzzy supra-uniformity to be any value in I = [0; 1], we introduce the concept of a gradation of supra-uniformity as follows:
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Deÿnition 3.1. Let D be the set of all maps u : I X → I X which satisÿes the above conditions (1)0 – (3)0 . A mapping U from D into I is said to be a gradation of supra-uniformity if it satisÿes the following axioms: (GSU1) ∃u ∈D s.t. U (u) = 1, (GSU2) U (u) = U (u−1 ); (GSU3) U (u) ¿ r; u 6 ⇒ U () ¿ r, and (GSU4) U (u) ¿ r ⇒ ∃∈D s.t. U () ¿ r; ◦ 6 u, = (()) where ◦ : I X → I X deÿned by ( ◦ )()V and u−1 : I X → I X deÿned by u−1 () = { ∈I X | u(1 − ) 6 1 − }. The pair (X; U ) is called a fuzzy supra-uniform space. Remark 3.1. If all grades of supra-uniformity are restricted to be 0 or 1, then Deÿnition 3.1 is reduced to the deÿnition of a (classical) fuzzy supra-uniform space. In other words, the new concept is a good extension for the (classical) fuzzy case. Proposition 3.1. Let (X; U ) be a fuzzy suprauniform space. Then, ∀r ∈(0; 1]; the family Ur = {u ∈D | U (u) ¿ r} is a (classical) fuzzy supra-uniformity on X . Proof. (SU1) From (GSU1) ∃u ∈D s.t. U (u) = 1. In other words ∃u ∈D s.t. U (u) ¿ r ∀r. Hence Ur 6= . (SU2) Let u ∈Ur . Then U (u−1 ) = U (u) ¿ r. Hence, u−1 ∈Ur . (SU3) Let u ∈Ur , u 6 . From (GSU3), we have U (u) ¿ r; u 6 . Thus, U () ¿ r. Consequently, ∈Ur . (SU4) Let u ∈Ur . Then, U (u) ¿ r. From (GSU 4), ∃ ∈D s.t. U () ¿ r; ◦ 6 u. Thus, ∃ ∈Ur ; ◦ 6 u. Hence, we have the result. Deÿnition 3.2. For each r ∈(0; 1], the family Ur is called the r-level (classical) fuzzy supra-uniformity of the gradation of supra-uniformity U . Proposition 3.2. Let (X; U ) be a fuzzy suprauniform space. The family {Ur : r ∈(0; 1]} of all r-level (classical) fuzzy supra-uniformities on X; w.r.t. U is a descending family. Furthermore, Ur = T U t. t¡r Proof. Let r ∈(0; 1]; t¡r andTu ∈Ur . Then, U (u) ¿ r¿t and u ∈Ut . Thus, Ur ⊂ t¡r Ut . Suppose that
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T there exists u ∈ t¡r Ut s.t. u 6∈ Ur . Then, U (u)¡r. This impliesT that ∃t¡r s.t. U (u)¡t¡r. Consequently, u 6∈ t¡r Ut ; this is a contradiction. Hence, we have the result. Proposition 3.3. Let (X; U ) be a (classical) fuzzy supra-uniform space. Then, the family U ; deÿned by i _ = { ∈I X | u() 6 for some u ∈U };
∈ U
is a (classical) fuzzy supra-topology on X . Proof. (1) Since u(1) ¿ 1 ∀u ∈U , then sup{ ∈I X |u() 6 1 for some u ∈U } = 1; i.e. 1 ∈ U . Since u(0) = 0 and u() ¿ ∀ ∈I X ; then ∈U } = 0, i.e. 0∈ U . sup{ ∈I X |u() 6 0 for some uW (2) Let ∈ U ∀ . Then, = {∈I X | u() 6 for some u ∈U }. Thus, __ _ = ∈I X | u() 6 for some u ∈U
= Thus,
_
W
) _ ∈ I u() 6 for some u ∈ U :
(
X
∈ U .
Deÿnition 3.3. Let (X; U ) be a (classical) fuzzy supra-uniform space. The family U , as deÿned above, is a (classical) fuzzy supra-topology induced by U . The pair (X; U ) is the (classical) fuzzy supratopological space induced by (X; U ). Proposition 3.4. Let (X; U ) be a fuzzy suprauniform space. For r ∈(0; 1]; let Ur be the (classical) fuzzy supra-topology induced by Ur . Then the familyT{ Ur | r ∈(0; 1]} is descending. Furthermore, Ur = s¡r Us . Proof. It is similar to that given in [1, Proposition 2.11] Proposition 3.5. Let (X; U ) be a fuzzy supraX uniform space. W The mapping U : I → I deÿned by U () = {r ∈(0; 1] | ∈ Ur } is a gradation of supra-openness on X . It is called the gradation of
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supra-openness induced by the gradation of suprauniformity U . Proof. It is similar to that given for [1, Proposition 2.12]. References [1] K.C. Chattopadhyay, R.H. Hazra, S.K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49 (1992) 236 –292.
[2] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. (58) 559 –571. [3] S.A. Aly El-Sheikh, Some strong properties of fuzzy bitopological spaces, Ph.D. Thesis, Ain Shams University, Faculty of Education, Egypt, 1995.