On smooth topological spaces IV

Fuzzy Sets and Systems 119 (2001) 473–482

www.elsevier.com/locate/fss

On smooth topological spaces IV A.A. Ramadan a ; ∗ , S.N. El-Deeb b , M.A. Abdel-Sattar a a

Department of Mathematics, Faculty of Science, Cairo University, Beni-Suef, Egypt Department of Mathematics, Faculty of Science, Cairo University, Fayium, Egypt

b

Received May 1998; received in revised form December 1998

Abstract We introduce the concept of convergence of nets in the framework of smooth topology and establish their fundamental c 2001 Elsevier Science B.V. All rights reserved. properties.
Keywords: Fuzzy nets; Smooth topology; Moore–Smith convergence; Fuzzifying topology

1. Introduction The concept of fuzzy topology was 6rst de6ned in 1968 by Chang [2] and later rede6ned in a somewhat di;erent way by Lowen [7] and by Hutton [6]. According to Sostak [11], in all these de6nitions, a 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 fuzzi6cation of the concept of the topological space. Therefore Sostak introduced a new de6nition of fuzzy topology in 1985 [11]. Later on, he developed the theory of fuzzy topological spaces in [11 – 13]. After that several authors [3,5,9] have introduced the smooth de6nition and studied smooth fuzzy topological spaces being unaware of Sostak’s works. In the present work, we shall study the convergence of fuzzy nets in smooth topological spaces in two ways, 6rst by using crisp points, second by using fuzzy points. 2. Notions and preliminaries The class of all fuzzy sets on a universe X will be denoted by I X where I = [0; 1], and fuzzy sets by capital letters as A; B; C, etc., and, the complement of a fuzzy set A of X will be denoted by Ac [15].

∗

Corresponding author.

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 0 8 3 - 4

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Denition 2.1 (Ying [14]). Let X be a nonempty set. Let x be a fuzzy point in X and let A be a fuzzy subset of X . Then the degree to which x belongs to A is m(x ; A) = min(1; 1 −  + A(x)). Obviously, we have the following properties: (1) m(x; A) = A(x); (2) m(x ; A)
= 1 i; x ∈ A; m(x ; A) = 0 i;  = 1 and A(x) = 0; (3) m(x ; t∈T At ) = supt∈T m(x ; At ) (generalized multiple choice principle). Denition 2.2 (Ying [14]). Let (X; ) be a fuzzy topological space (fts, for short), let e be a fuzzy point in X and let A be a fuzzy subset of X . Then the degree to which A is a neighborhood of e (nbd, for short) is de6ned by Ne∗ (A) = sup{m(e; B): B ∈ and B ⊆ A}: Thus Ne∗ ∈ I (I

x

)

is called the fuzzy nbd system of e in (X; ).

Denition 2.3 (Pu and Liu [8]). Let (D; ¿) be a directed set. Let X be an ordinary set. Let M be the collection of all fuzzy points in X . The function S : D → M is called a fuzzy net in X . In other words, a fuzzy net is a pair (S; ¿) such that S is a function from D into M and ¿ directs the domain of S. For n ∈ D; S(n) is often denoted by Sn . Denition 2.4 (Ying [14]). Let S be a fuzzy net in a fts (X; ), and e a fuzzy point in X . Then the degree to which S converges to e and S clusters to e are given, respectively, as follows: Con∗ (S; e) = inf {1 − Ne∗ (A): S is often in Ac }; Cl∗ (S; e) = inf {1 − Ne∗ (A): S is 6nally in Ac }: Denition 2.5 (Ying [14]). Let (X; ) be a fts, e a fuzzy point in X and A a fuzzy subset of X . Then the degree to which e is an adherent point of A is given as ad(e; A) = inf c (1 − Ne∗ (B)): B⊆A

Proposition 2.1 (Ying [14]). Let (X; ) and (Y; ) be two fts’s and f a mapping from X into Y . Then f is a fuzzy continuous function i4 for any fuzzy point e in X and any fuzzy subset V of Y; ∗Y Nf(e) (V )6 sup{Ne∗X (U ): u ∈ I X ; f(U ) ⊆ V }:

Denition 2.6 (Ramadan [9]). A smooth topological space (sts, for short), is an ordered pair (X; ) where X is a nonempty set and : I X → I is a mapping satisfying the following properties: (01) (0) = (1) = 1, (02) ∀A1 ; A2
∈ I X ;  (A1 ∩ A2 )¿ (A1 ) ∧ (A2 ), (03) ∀I; i∈I Ai ¿ inf i∈I (Ai ). Denition 2.7 (Ramadan [9]). A mapping f : X → Y is called smooth continuous with respect to the smooth topologies on X and on Y i; for every A ∈ I Y , we have (f−1 (A))¿ (A), where f−1 (A) is de6ned by f−1 (A)(x) = A(f(x)). Denition 2.8 (Ramadan [9]). Let (X; ) be a sts, and  ∈ (0; 1], then the family  = {A ∈ I X : (A)¿} which is clearly a fuzzy topology in Chang’s sense [3] is called the “-cut”.

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Denition 2.9 (Ramadan et al. [10]). Let (X; ) be sts, let x be a crisp point in X , and let A be a fuzzy subset of X . Then the degree to which A is a nbd of x is de6ned by Nx (A) = sup{B(x) ∧ (B): B ⊆ A and (B) ¿ 0}. Denition 2.10 (Ramadan et al. [10]). The closure AH of A is de6ned as follows: H = inf {(1 − (Bc )) ∨ B(x)}: A(x) A⊆B

Theorem 2.1 (Ramadan et al. [10]). Let (X; ) be a sts; x a crisp point in X; and A a fuzzy subset of X . Then; (1) A0 (x) = supB⊆A Nx (B); H c (x) = 1 − ad(x; A); (2) (A) H (3) A ⊆ A; H (4) A ⊆ B ⇒ AH ⊆ B: Denition 2.11 (Ramadan et al. [10]). Let (X; ) be a sts, e a fuzzy point in X , and A be a fuzzy subset of X . Then the degree to which A is a nbd of e is de6ned as Ne (A) = sup {m(e; B) ∧ (B): (B) ¿ 0}: B⊆A X

Thus Ne ∈ (I )I is called the smooth nbd system of e in (X; ). Lemma 2.2 (Ramadan et al. [10]). sup sup{A(x) ∧ B(x): A(x)¿} = sup sup{ ∧ B(x): A(x)¿}:

∈[0;1]

∈[0;1]

Denition 2.12. Let (X; ) be a sts, x a crisp point in X; and A a fuzzy subset of X . Then the degree to which x is an adherent point of A is given as ad(x; A) = inf c (1 − Nx (B)): B⊆A

Denition 2.13 (Ramadan et al. [10]). Let (X; ) be a sts, e a fuzzy point in X and A a fuzzy subset of X . Then the degree to which e is an adherent point of A is given as, ad(e; A) = inf B⊆Ac (1 − Ne (B)). Proposition 2.2 (Ramadan et al. [10]). The nbd systems Nx can be constructed from the cuts  ;  ¿ 0 by using the equality Nx (A) = sup¿0 {[Nx∗ (A)] ∧ }; where [Nx∗ (A)] = supB⊆A {B(x): B ∈  }. Proposition 2.3 (Ramadan et al. [10]). The nbd systems Ne of e in sts can be constructed by Ne (A) = sup {[Ne∗ (A)] ∧ }; ¿0

where [Ne∗ (A)] = sup{m(e; B): B ⊆ A and B ∈  }:

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3. Convergence of nets in smooth topology by using crisp points Denition 3.1. Let S be a fuzzy net in a sts (X; ), and x a crisp point in X . Then the degree to which S converges to x and S clusters to x are given, respectively, as follows: Con(S; x) = inf {1 − Nx (A): S is often in Ac }; Cl(S; x) = inf {1 − Nx (A): S is 6nal in Ac }: Proposition 3.1. Let (X; ) be a sts; and let S be a fuzzy net in X . Then for any crisp point x in X; we have (i) Con(S; x) = inf ¿0 {Con∗ (S; x) ∨ (1 − )}; (ii) Cl(S; x) = inf ¿0 {Cl∗ (S; x) ∨ (1 − )}; where Con∗ (S; x) = inf {1 − [Nx∗ (A)] : S is often in Ac }; Cl∗ (S; x) = inf {1 − [Nx∗ (A)] : S is 6nal in Ac }; depends on . Proof. (i) From De6nition 3.1, we have Con(S; x) = inf {1 − Nx (A): S is often in Ac }   ∗  c = inf 1 − sup{[Nx (A)] ∧ }: S is often in A ¿0

(by Proposition 2:2)

= inf inf {(1 − [Nx∗ (A)] ) ∨ (1 − ): S is often in Ac } ¿0

= inf {inf {(1 − [Nx∗ (A)] ): S is often in Ac } ∨ (1 − )} ¿0

= inf {Con∗ (S; x) ∨ (1 − )}: ¿0

(ii) Similar to (i). Lemma 3.1. Let S be a fuzzy net in fts (X; ) and x a crisp point in X . Then Cl∗ (S; x) = sup{Con∗ (T; x): T is a fuzzy subset of S}: Proof. Similar to Theorem 3.3 in [14]. Lemma 3.2. Let (X; ) be a fts; x a crisp point in X and A a fuzzy subset of X . Then ad ∗ (x; A) = sup{Con∗ (S; x): S is a fuzzy net in A}; where ad ∗ (x; A) = inf c (1 − [Nx∗ (B)] ): B⊆A

Proof. Similar to Theorem 3.1 in [14]. Theorem 3.1. Let S be a fuzzy net in a sts (X; ) and x a crisp point in X . Then Cl(S; x)¿ sup{Con(T; x): T is a fuzzy subset of S}:

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Proof. From Proposition 3.1(ii) Cl(S; x) = inf {Cl∗ (S; x) ∨ (1 − )}:

(1)

¿0

From Lemma 3.1, Cl∗ (S; x) = sup{Con∗ (T; x): T is a fuzzy subset of S}:

(2)

Using (2) in (1) and by Proposition 3.1(i) we have Cl(S; x) = inf {sup{Con∗ (T; x): T is a fuzzy subset of S} ∨ (1 − )} ¿0   ¿ sup

inf {Con∗ (T; x) ∨ (1 − )}: T is a fuzzy subset of S

¿0

= sup{Con(T; x): T is a fuzzy subset of S}: Proposition 3.2. Let (X; ) be a sts; x a crisp point of X and A a fuzzy subset of X. Then ad(x; A) = inf {ad ∗ (x; A) ∨ (1 − )}: ¿0

Proof. From De6nition 2.12 ad(x; A) = inf c (1 − Nx (B)) B⊆A   ∗  = inf c 1 − sup ([Nx (A)] ∧ ) B⊆A

¿0

(by Proposition 2:3)

= inf c inf {(1 − [Nx∗ (A)] ) ∨ (1 − )} B⊆A ¿0   = inf inf c (1 − [Nx∗ (A)] ) ∨ (1 − ) ¿0

B⊆A

= inf {ad ∗ (x; A) ∨ (1 − )} (by Lemma 3:2): ¿0

Theorem 3.2. Let (X; ) be a sts; x a crisp point in X and A ∈ I X . Then ad(x; A)¿ sup{Con(S; x): S is a fuzzy net in A}: Proof. From Proposition 3.1(i) Con(S; x) = inf {Con∗ (S; x) ∨ (1 − )}: ¿0

(3)

From Lemma 3.2 ad ∗ (x; A) = sup{Con∗ (S; x): S is a fuzzy net in A}: Now, using (3) and (4) we have sup{Con(S; x): S is a fuzzy net in A} = sup inf {Con∗ (S; x) ∨ (1 − ): S is a fuzzy net in A} ¿0

6 inf {ad ∗ (x; A) ∨ (1 − )} ¿0

= ad(x; A)

(by Proposition 3:2):

(4)

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Theorem 3.3. Let (X; ) be a sts; x a crisp point and S a fuzzy net in X. For any n ∈ D; let An = Then C1(S; x) = inf n∈D ad(x; An ).


n6m

Sm .

Proof. S is 6nally in Bc ⇔ ∃n ∈ D such that Sm ∈ Bc ; ∀m¿n ⇔ ∃n ∈ D such that An ⊆ Bc ; ⇔ ∃n ∈ D such that B ⊆ Acn : Therefore, inf ad(x; An ) = inf inf c (1 − Nx (B))

n∈D

n∈D B⊆An

= inf {1 − Nx (B): ∃n ∈ D such that B ⊆ Acn } = C1(S; x): 4. Convergence of nets in smooth topology by using fuzzy points Denition 4.1. Let S be a fuzzy net in a sts (X; ), and e a fuzzy point in X . Then the degree to which S converges to e and S clusters to e are given, respectively, as follows: Con(S; e) = inf {1 − Ne (A): S is often in Ac }; Cl(S; e) = inf {1 − Ne (A): S is 6nal in Ac }: Theorem 4.1. Let (X; ) be a sts; and let S be a fuzzy net in X; e a fuzzy point in X;  ∈ (0; 1]. Then the degree to which S converges to e and S cluster to e are given; respectively; as follows: (i) Con(S; e) = inf ¿0 {Con∗ (S; e) ∨ (1 − )}; (ii) Con(S; x ) = inf ¿0 {max(0;  + Con∗ (S; x) − 1) ∨ (1 − )}; (iii) Cl(S; e) = inf ¿0 {Cl∗ (S; e) ∨ (1 − )}; (iv) Cl(S; x ) = inf ¿0 {max(0;  + Cl∗ (S; x) − 1) ∨ (1 − )}; where Con∗ (S; e) = inf {1 − [Ne∗ (A)] : S is often in Ac }; Cl∗ (S; x) = inf {1 − [Ne∗ (A)] : S is often in Ac }: Proof. (i) and (iii) similar to Proposition 3.1. Now we prove (ii), Con(S; x ) = inf {Con∗ (S; x ) ∨ (1 − )} ¿0

= inf {inf {1 − [Nx∗ (A)] : S is often in Ac } ∨ (1 − )} ¿0

= inf {inf {1 − {min(1; 1 −  + [Nx∗ (A)] )}: S is often in Ac } ∨ (1 − )} ¿0

(Theorem 3:1 in [10]) = inf {inf {max(0;  − [Nx∗ (A)] ): S is often in Ac } ∨ (1 − )} ¿0

= inf {inf {max(0;  + 1 − [Nx∗ (A)] − 1): S is often in Ac } ∨ (1 − )} ¿0

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479

= inf {max(0;  + inf {1 − [Nx∗ (A)] : S is often in Ac } − 1) ∨ (1 − )} ¿0

= inf {max(0;  + Con∗ (S; x) − 1) ∨ (1 − )}: ¿0

(iv) Similar to (ii). Proposition 4.1. Let (X; ) be a sts; e a fuzzy point in X; and A a fuzzy subset in X. Then ad(e; A) = inf {ad ∗ (e; A) ∨ (1 − )}; ¿0

where ad ∗ (e; A) = inf c (1 − [Ne∗ (B)] ): B⊆A

Proof. Similar to Proposition 3.2. Lemma 4.1. Let S be a fuzzy net in fts (X; ) and e a fuzzy point in X. Then; Cl∗ (S; e) = sup{Con∗ (T; e): T is a fuzzy subset of S}: Proof. Similar to Theorem 3.3 in [14]. Lemma 4.2. Let (X; ) be a fts; e a fuzzy point in X and A a fuzzy subset of X. Then; ad ∗ (e; A) = sup{Con∗ (S; e): S is a fuzzy net in A}: Proof. Similar to Theorem 3.1 in [14]. Theorem 4.2. Let (X; ) be a sts; e a fuzzy point in X and for any A ∈ I X . Then; ad(e; A)¿ sup{Con(S; e): S is a fuzzy net in A}: Proof. From Proposition 4:1(i) Con(S; e) = inf {Con∗ (S; e) ∨ (1 − )}: ¿0

(5)

From Lemma 4.2, ad ∗ (e; A) = sup{Con∗ (S; e): S is a fuzzy net in A}: Now, using (5) and (6) we have sup{Con(S; e): S is a fuzzy net in A} = sup inf {Con∗ (S; e) ∨ (1 − ): S is a fuzzy net in A} ¡0

6 inf {ad ∗ (e; A) ∨ (1 − )} ¿0

= ad(e; A)

(by Proposition 4:2):

(6)

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Theorem 4.3. Let S be a fuzzy net in a sts (X; ) and e a fuzzy point in X. Then Cl(S; e)¿ sup{Con(T; e): T is a fuzzy subet of S}: Proof. From Proposition 4.1(ii) Cl(S; e) = inf {Cl∗ (S; e) ∨ (1 − )}:

(7)

¿0

From Lemma 4.1, Cl∗ (S; e) = sup{Con∗ (T; e): T is a fuzzy subset of S}:

(8)

Using (8), (7) and Proposition 4.1(i) we have Cl(S; e) = inf {sup{Con∗ (T; e): T is a fuzzy subset of S} ∨ (1 − )} ¿0   ¿ sup inf {Con∗ (T; e) ∨ (1 − )}: T is a fuzzy subset of S ¿0

= sup{Con(T; x): T is a fuzzy subset of S}: Theorem 4.4. Let (X; ) be a sts, e a fuzzy point and S a fuzzy net in X . For any n ∈ D, let An = Then,


n6m

Sm .

Cl(S; e) = inf ad(e; An ): n∈D

Proof. Similar to Theorem 3.2. Lemma 4.3. Let (X; ) and (Y; ) be two fts’s and f a mapping from X into Y . Then f is a fuzzy continuous function i4 for any fuzzy point e in X and any fuzzy subset V of Y , Y [Nf∗(e) (V )] 6 sup{[Ne∗ X (U )] : U ∈ I X ; f(U ) ⊆ V }:

Proof. Similar to Proposition 2.2 in [14]. Theorem 4.5. Let f : (X; ) → (Y; ) be a smooth continuous mapping, where and the following statements hold: (1) For any fuzzy net S and fuzzy point e in X ,

are two sts’s. Then

Con(S; e)6Con(f ◦ S; f(e)): H ⊆ f(A). (2) For every A ∈ I X ; f(A) H (3) For every B ∈ I Y ; f−1 (B) ⊆ f−1 (B). (4) For every B ∈ I Y ; f−1 (B0 ) ⊆ (f−1 (B))0 . Proof. (1) Let f : X → Y be a smooth continuous function with respect to and that for any fuzzy net S and fuzzy point e in X , inf {Con∗ (S; e) ∨ (1 − )}6 inf {Con∗ (f ◦ S; f(e)) ∨ (1 − )}:

¿0

¿0

. Then, we want to prove

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481

By Proposition 4.2 in [3], and Lemma 4.3, it suKces to show that, Con∗ (S; e)6Con∗ (f ◦ S; f(e)), thus Y Con∗ (f ◦ S; f(e)) = inf {1 − [Nf∗(e) (B)] : f ◦ S is often in Bc }

¿ inf {1 − sup{[Ne∗ X (A)] : f(A) ⊆ B}: f ◦ S is often in Bc } = inf {1 − [Ne∗ X (A)] : ∃B ∈ I Y such that f(A) ⊆ B and f ◦ S is often in Bc }: If f(A) ⊆ B and f ◦ S is often in Bc , i.e., for any n ∈ D there exists p ∈ D such that p¿n and f(Sp ) ∈ Bc . Let Sp = x . Then f(x ) = f(Sp ) ∈ Bc , i.e., 61 − B(f(x))61 − f(A)(f(x))61 − A(x); Sp ∈ Ac . Therefore, S is often in Ac and Con∗ (f ◦ S; f(e))¿ inf {1 − [Ne∗ X (A)] : S is often in Ac } = Con∗ (S; e) and so, Con∗ (f ◦ S; f(e)) ∨ (1 − )¿Con∗ (S; e) ∨ (1 − ); therefore inf {Con∗ (f ◦ S; f(e)) ∨ (1 − )}¿ inf {Con∗ (S; e) ∨ (1 − )};

¿0

¿0

i.e. Con(f ◦ S; f(e))¿Con(S; e): (2) For any fuzzy subset A of X and x ∈ X , using (1) we get H c ) = 1 − (A) H c (x) = A(x) H ad(x; A) = 1 − m(x; (A)

(by Theorem 2:1(2)):

Hence, for any y ∈ Y , it follows that: H H = sup ad(x; A) f(A)(y) = sup A(x) f(x)=y

f(x)=y

= sup sup{Con(S; x): S is a fuzzy net in A} f(x)=y

= sup sup inf {Con∗ (S; x) ∨ (1 − ): S is a fuzzy net in A} f(x)=y

¿0

6 sup sup inf {Con∗ (f ◦ S; f(x)) ∨ (1 − ): S is a fuzzy net in A} f(x)=y

¿0

= sup inf {Con∗ (f ◦ S; y) ∨ (1 − ): S is a fuzzy net in A} ¿0

6 sup inf {Con∗ (T; y) ∨ (1 − ): T is a fuzzy net in f(A)} ¿0

H ⊆ f(A): = ad(y; f(A)) = f(A)(y); i:e: f(A) (3) For every B ∈ I Y , we get from (2), H f(f−1 (B)) ⊆ f(f−1 (B)) ⊆ BH ⇒ f−1 (B) ⊆ f−1 f(f−1 (B)) ⊆ f−1 (B); H i.e. f−1 (B) ⊆ f−1 (B).

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(4) For every B ∈ I Y , using (3) and Theorem 2.1, we observe that f−1 (Bc ) = f−1 ((B0 )c ) = [f−1 (B0 )]c ⊇ f−1 (Bc ) = (f−1 (B))c = [f−1 (B)0 ]c : Hence f−1 (B0 ) ⊆ [f−1 (B)]0 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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