- Email: [email protected]
Quotient fuzzy topological spaces
Fuzzy Sets and Systems 119 (2001) 543–545
www.elsevier.com/locate/fss
Quotient fuzzy topological spaces Francisco Gallego Lupi&an˜ ez ∗ Departamento de Geometra & y Topologia, Facultad de Matematicas, Universidad Complutense, 28040-Madrid, Spain Received September 1997; received in revised form October 1998
Abstract In this paper we study properties of certain fuzzy maps associated to a map, and we apply these properties to obtain results c 2001 Elsevier Science B.V. All rights reserved. on quotient fuzzy topological spaces. MSC: primary 54A40; secondary 04A72 Keywords: F-continuous map; Quotient fuzzy topological space
1. Introduction and denitions Let Top be the category of all the topological spaces and the continuous maps and let CFT be the category of the fuzzy topological spaces in the Chang’s sense and the F-continuous maps. We will denote ! : Top → CFT to be the functor which associates to any topological space (X; ), the fuzzy space (X; !()) where !() is the totality of all lower semicontinuous maps of (X; ) to the unit interval. It is called the weakly induced fuzzy topological space by (X; ) [3]. In this paper we study properties of certain fuzzy ˜ where f : (X; ) → (Y; s) is a continuous map. maps f, In previous papers [4 – 6] we have applied this fuzzy map to study invariance of fuzzy properties, now we apply these properties to obtain results on quotient fuzzy topological spaces. Next, we list the deAnitions which we will use in this paper.
∗
Tel.: + 34-1-394-4566; fax: + 34-1-394-4607. E-mail address: [email protected] (F.G. Lupi&an˜ ez)
Denition 1 (Chang [2]). Let X and Y be two sets and let f be a map from X to Y . Let be a fuzzy set in Y , then the inverse of written as f−1 ( ), is deAned by f−1 ( )(x) = (f(x)) for all x in X . Conversely, if is a fuzzy set in X , the image of , written as f( ), is a fuzzy set in Y given by f( )(y) =
supx∈f−1 (y) { (x)} 0
if f−1 (y) = ∅; otherwise:
Denition 2 (Pu and Liu [7]). A fuzzy set in X is called a fuzzy point if it takes the value 0 for all y ∈ X except one, say, x ∈ X . If this value at x is (0 ¡ 61) we denote this fuzzy point by x . Denition 3 (Lupi&an˜ ez [4]). Let X and Y be two sets and f : X → Y be a map. We denote f˜ the map given ˜ ) = f(x ) for each fuzzy point x in X . by f(x Denition 4 (Lowen [3]). A fuzzy extension of a topological property is said to be good, when it is possessed by (X; !()) if, and only if, the original property is possessed by (X; ).
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 4 0 - 8
F.G. Lupi&an˜ ez / Fuzzy Sets and Systems 119 (2001) 543–545
544
Denition 5 (Wong [9]). If (X; T ) is a fuzzy topological space and R is an equivalence relation deAned on X , we denote by p the usual projection from X onto X=R. Let S be the family of fuzzy sets in X=R deAned by S = { | p−1 ( ) ∈ T }, then S is called the quotient topology for X=R. (X=R; S) is called the quotient fuzzy topological space, and p will be called the quotient fuzzy map.
Proof. (a) For each y fuzzy point in Y , we have y ∈ Y , then there exists at least an x ∈ X such that ˜ ) = y . f(x) = y and f(x (b) If x ; x are two fuzzy points in X such that ˜ ), then, by Remark 2.1, is f(x) = f(x ) ˜ f(x ) = f(x (⇒ x = x ) and = . Thus x = x .
2. Properties of the fuzzy map associated to a map
Proof. For each fuzzy point in y in Y , by the hypothesis, then there uniquely exists an x ∈ X , such that −1 (y ) = x . f(x) = y, i.e. f(y) = x. Now we prove f −1 Otherwise, let f (y ) = x ; = , by Remark 2.1, −1 (y )). This is a contradiction. ˜ ) = f( ˜f we have f(x −1 Therefore f (y ) is uniquely the fuzzy point in X which takes the value in f−1 (y).
Remark 2.1. If X and Y are two sets, and f : X → Y is an onto map, then for each fuzzy point x in X we ˜ ) is the fuzzy point in Y that takes the have that f(x value in f(x). Proof. ˜ )(y) = f(x =
0
sup {x (t)}
t ∈ f−1 (y)
if x ∈ f−1 (y) (i:e: f(x) = y); if x ∈= f−1 (y) (i:e: f(x) = y):
˜ ) is the fuzzy point in Y that takes the value Then, f(x in f(x). Proposition 2.2. Let X; Y; Z be three sets; and f : X → Y and g : Y → Z be two onto maps; then ˜ g] ◦ f = g˜ ◦ f. Proof. For each fuzzy point x in X , we have that ˜ ) is the fuzzy point in Y that takes the value f(x ˜ )) is the fuzzy point in Z in f(x), and g( ˜ f(x that takes the value in g(f(x)) = (g ◦ f)(x). Then, ˜ )) = (g] g( ˜ f(x ◦ f)(x ) for each fuzzy point x , and, ˜ ] g ◦ f = g˜ ◦ f. Remark 2.3. Let X be a set, and f : X → X an onto map. If f˜ is the identity, then f is also the identity. ˜ ) = x is the fuzzy point which takes the Proof. f(x value in f(x), then f(x) = x, for each x ∈ X . Proposition 2.4. Let X; Y; be two sets. (a) If f : X → Y be an onto map; then f˜ is also onto. (b) If f : X → Y be a one-to-one map; then f˜ is oneto-one.
Proposition 2.5. Let X; Y be two sets; and f be a −1 = f˜−1 . one-to-one map from X to Y; then f
f˜−1 (y )(x) = y (f(x)) if f(x) = y (⇔ x = f−1 (y)); = 0 if f(x) = y: −1 (y ). Then f˜−1 (y ) = f Proposition 2.6. Let X; Y; be two sets; and f : X → Y a map. (a) If f˜ is onto then f is also onto. (b) If f˜ is one-to-one; then f is also one-to-one. Proof. (a) For each y ∈ Y , let y1 be the fuzzy point in Y which takes the value 1 in y. By the hypothesis, ˜ ) = y1 . there exists a fuzzy point x in X such that f(x ˜ )(y) and f−1 (y) = ∅. Then 1 = f(x (b) If there exist t; t ∈ X with f(t) = f(t ) = y and t = t , let t1 ; t1 be the fuzzy points in X which takes the value 1 in t; t . We have that 1=
˜ 1 )(y) and f(t ˜ 1 )(y ) = 0 sup {t1 (x)} = f(t
x ∈ f−1 (y)
for all y = y ˜ 1 ) = f(t ˜ ) and and analogously for t1 . Then f(t 1 t1 = t1 . Proposition 2.7. Let (X; ); (Y; s) be two topological spaces. If f : (X; ) → (Y; s) is continuous; then f˜ : (X; !()) → (Y; !(s)) is F-continuous.
F.G. Lupi&an˜ ez / Fuzzy Sets and Systems 119 (2001) 543–545
Proof. For each open fuzzy set in (Y; !(s)) we have that −1 ((a; 1]) ∈ s for all a ∈ [0; 1), and by the hypothesis, f−1 ( −1 ((a; 1])) ∈ . Then ◦ f is open fuzzy set in (X; !()). Finally, f˜−1 ( ) is open fuzzy set in (X; !()) because f˜−1 ( )(x) = (f(x)). Remark 2.8. The converse of the above proposition is also true [4]. 3. On quotient fuzzy topological spaces Proposition 3.1. Let (X; ); (Y; s) be two topological spaces. Then f˜ : (X; !()) → (Y; !(s)) is a quotient fuzzy map if; and only if, f : (X; ) → (Y; s) is a quotient map. Proof. (⇒) We have that A ∈ s if, and only if A is open fuzzy set in (Y; !(s)). This is equivalent to f˜−1 ( A ) = f−1 (A) is an open fuzzy set in (X; !()), and f−1 (A) ∈ . (⇐) We have that is open fuzzy set in (Y; !(s)) if, and only if, −1 ((a; 1]) ∈ s for each a ∈ [0; 1). By the hypothesis, this is equivalent to f−1 ( −1 ((a; 1])) ∈ for each a ∈ [0; 1), i.e. f˜−1 ( ) is open fuzzy set in (X; !()). Corollary 3.2. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ : (X; !()) → (Y; !(s)) be a quotient fuzzy map. If : (a) (X; !()) veri7es the fuzzy version of a property P of topological spaces; (b) the property P is invariant under quotient maps; (c) the fuzzy version of P is a good extension of P. Then (Y; !(s)) veri7es the fuzzy version of P. Corollary 3.3. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ : (X; !()) → (Y; !(s)) be a quotient fuzzy map. If : (a) (Y; !(s)) veri7es the fuzzy version of a property P of topological spaces; (b) the property P is inverse invariant under quotient maps; (c) the fuzzy version of P is a good extension of P. Then (X; !()) veri7es the fuzzy version of P.
545
Proposition 3.4. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ be a F-continuous map from (X; !()) onto (Y; !(s)). If there exist a F-continuous map s˜ from (Y; !(s)) to (X; !()) such that f˜ ◦ s˜ = 1Y ; then (Y; !(s)) is F-homeomorphic with X=R; where R is the equivalence relation x1 R x2 ⇔ f(x1 ) = f(x2 ) and the fuzzy topology is the quotient. Proof. We have f˜ ◦ s˜ = 1Y then, by Proposition 2.2 and Remark 2.3, f ◦ s = 1Y . Then, the map g : X=R → Y induced by f is a homeomorphism. Finally, by Propositions 2.4, 2.7 and 2.5, g˜ is onto, one-to-one, F-continuous, and g˜−1 is also F-continuous. Remark 3.5. Using this fuzzy maps, we can generalize to Fuzzy Topology, other results concerning quotient spaces. References & ements de Math&ematique, Topologie G&en&eral, [1] N. Bourbaki, El& Hermann, Paris, 1971. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182 –190. [3] R. Lowen, A comparison of diMerent compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446 – 454. [4] F.G. Lupi&an˜ ez, Fuzzy perfect maps and fuzzy paracompactness, Fuzzy Sets and Systems 98 (1998) 137 –140. [5] F.G. Lupi&an˜ ez, Invariance of fuzzy properties, Mat. Vesnik 50 (1998) 11–16. [6] F.G. Lupi&an˜ ez, Finite-to-one fuzzy maps and fuzzy perfect maps, Kybernetika 34 (1998) 163 –169. [7] P.-M. Pu, Y.-M. Liu, Fuzzy topology I. Neighbourhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571– 599. [8] P.-M. Pu, Y.-M. Liu, Fuzzy topology II. Product and quotient spaces, J. Math. Anal. Appl. 77 (1980) 20 – 37. [9] C.K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45 (1974) 512 – 521. [10] C.K. Wong, Fuzzy topology in fuzzy sets and their applications to cognitive and decision processes, Proc. US –Japan Sem. Univ. California, Berkeley, 1974, Academic Press, New York, 1975, pp. 171–190.
www.elsevier.com/locate/fss
Quotient fuzzy topological spaces Francisco Gallego Lupi&an˜ ez ∗ Departamento de Geometra & y Topologia, Facultad de Matematicas, Universidad Complutense, 28040-Madrid, Spain Received September 1997; received in revised form October 1998
Abstract In this paper we study properties of certain fuzzy maps associated to a map, and we apply these properties to obtain results c 2001 Elsevier Science B.V. All rights reserved. on quotient fuzzy topological spaces. MSC: primary 54A40; secondary 04A72 Keywords: F-continuous map; Quotient fuzzy topological space
1. Introduction and denitions Let Top be the category of all the topological spaces and the continuous maps and let CFT be the category of the fuzzy topological spaces in the Chang’s sense and the F-continuous maps. We will denote ! : Top → CFT to be the functor which associates to any topological space (X; ), the fuzzy space (X; !()) where !() is the totality of all lower semicontinuous maps of (X; ) to the unit interval. It is called the weakly induced fuzzy topological space by (X; ) [3]. In this paper we study properties of certain fuzzy ˜ where f : (X; ) → (Y; s) is a continuous map. maps f, In previous papers [4 – 6] we have applied this fuzzy map to study invariance of fuzzy properties, now we apply these properties to obtain results on quotient fuzzy topological spaces. Next, we list the deAnitions which we will use in this paper.
∗
Tel.: + 34-1-394-4566; fax: + 34-1-394-4607. E-mail address: [email protected] (F.G. Lupi&an˜ ez)
Denition 1 (Chang [2]). Let X and Y be two sets and let f be a map from X to Y . Let be a fuzzy set in Y , then the inverse of written as f−1 ( ), is deAned by f−1 ( )(x) = (f(x)) for all x in X . Conversely, if is a fuzzy set in X , the image of , written as f( ), is a fuzzy set in Y given by f( )(y) =
supx∈f−1 (y) { (x)} 0
if f−1 (y) = ∅; otherwise:
Denition 2 (Pu and Liu [7]). A fuzzy set in X is called a fuzzy point if it takes the value 0 for all y ∈ X except one, say, x ∈ X . If this value at x is (0 ¡ 61) we denote this fuzzy point by x . Denition 3 (Lupi&an˜ ez [4]). Let X and Y be two sets and f : X → Y be a map. We denote f˜ the map given ˜ ) = f(x ) for each fuzzy point x in X . by f(x Denition 4 (Lowen [3]). A fuzzy extension of a topological property is said to be good, when it is possessed by (X; !()) if, and only if, the original property is possessed by (X; ).
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 4 0 - 8
F.G. Lupi&an˜ ez / Fuzzy Sets and Systems 119 (2001) 543–545
544
Denition 5 (Wong [9]). If (X; T ) is a fuzzy topological space and R is an equivalence relation deAned on X , we denote by p the usual projection from X onto X=R. Let S be the family of fuzzy sets in X=R deAned by S = { | p−1 ( ) ∈ T }, then S is called the quotient topology for X=R. (X=R; S) is called the quotient fuzzy topological space, and p will be called the quotient fuzzy map.
Proof. (a) For each y fuzzy point in Y , we have y ∈ Y , then there exists at least an x ∈ X such that ˜ ) = y . f(x) = y and f(x (b) If x ; x are two fuzzy points in X such that ˜ ), then, by Remark 2.1, is f(x) = f(x ) ˜ f(x ) = f(x (⇒ x = x ) and = . Thus x = x .
2. Properties of the fuzzy map associated to a map
Proof. For each fuzzy point in y in Y , by the hypothesis, then there uniquely exists an x ∈ X , such that −1 (y ) = x . f(x) = y, i.e. f(y) = x. Now we prove f −1 Otherwise, let f (y ) = x ; = , by Remark 2.1, −1 (y )). This is a contradiction. ˜ ) = f( ˜f we have f(x −1 Therefore f (y ) is uniquely the fuzzy point in X which takes the value in f−1 (y).
Remark 2.1. If X and Y are two sets, and f : X → Y is an onto map, then for each fuzzy point x in X we ˜ ) is the fuzzy point in Y that takes the have that f(x value in f(x). Proof. ˜ )(y) = f(x =
0
sup {x (t)}
t ∈ f−1 (y)
if x ∈ f−1 (y) (i:e: f(x) = y); if x ∈= f−1 (y) (i:e: f(x) = y):
˜ ) is the fuzzy point in Y that takes the value Then, f(x in f(x). Proposition 2.2. Let X; Y; Z be three sets; and f : X → Y and g : Y → Z be two onto maps; then ˜ g] ◦ f = g˜ ◦ f. Proof. For each fuzzy point x in X , we have that ˜ ) is the fuzzy point in Y that takes the value f(x ˜ )) is the fuzzy point in Z in f(x), and g( ˜ f(x that takes the value in g(f(x)) = (g ◦ f)(x). Then, ˜ )) = (g] g( ˜ f(x ◦ f)(x ) for each fuzzy point x , and, ˜ ] g ◦ f = g˜ ◦ f. Remark 2.3. Let X be a set, and f : X → X an onto map. If f˜ is the identity, then f is also the identity. ˜ ) = x is the fuzzy point which takes the Proof. f(x value in f(x), then f(x) = x, for each x ∈ X . Proposition 2.4. Let X; Y; be two sets. (a) If f : X → Y be an onto map; then f˜ is also onto. (b) If f : X → Y be a one-to-one map; then f˜ is oneto-one.
Proposition 2.5. Let X; Y be two sets; and f be a −1 = f˜−1 . one-to-one map from X to Y; then f
f˜−1 (y )(x) = y (f(x)) if f(x) = y (⇔ x = f−1 (y)); = 0 if f(x) = y: −1 (y ). Then f˜−1 (y ) = f Proposition 2.6. Let X; Y; be two sets; and f : X → Y a map. (a) If f˜ is onto then f is also onto. (b) If f˜ is one-to-one; then f is also one-to-one. Proof. (a) For each y ∈ Y , let y1 be the fuzzy point in Y which takes the value 1 in y. By the hypothesis, ˜ ) = y1 . there exists a fuzzy point x in X such that f(x ˜ )(y) and f−1 (y) = ∅. Then 1 = f(x (b) If there exist t; t ∈ X with f(t) = f(t ) = y and t = t , let t1 ; t1 be the fuzzy points in X which takes the value 1 in t; t . We have that 1=
˜ 1 )(y) and f(t ˜ 1 )(y ) = 0 sup {t1 (x)} = f(t
x ∈ f−1 (y)
for all y = y ˜ 1 ) = f(t ˜ ) and and analogously for t1 . Then f(t 1 t1 = t1 . Proposition 2.7. Let (X; ); (Y; s) be two topological spaces. If f : (X; ) → (Y; s) is continuous; then f˜ : (X; !()) → (Y; !(s)) is F-continuous.
F.G. Lupi&an˜ ez / Fuzzy Sets and Systems 119 (2001) 543–545
Proof. For each open fuzzy set in (Y; !(s)) we have that −1 ((a; 1]) ∈ s for all a ∈ [0; 1), and by the hypothesis, f−1 ( −1 ((a; 1])) ∈ . Then ◦ f is open fuzzy set in (X; !()). Finally, f˜−1 ( ) is open fuzzy set in (X; !()) because f˜−1 ( )(x) = (f(x)). Remark 2.8. The converse of the above proposition is also true [4]. 3. On quotient fuzzy topological spaces Proposition 3.1. Let (X; ); (Y; s) be two topological spaces. Then f˜ : (X; !()) → (Y; !(s)) is a quotient fuzzy map if; and only if, f : (X; ) → (Y; s) is a quotient map. Proof. (⇒) We have that A ∈ s if, and only if A is open fuzzy set in (Y; !(s)). This is equivalent to f˜−1 ( A ) = f−1 (A) is an open fuzzy set in (X; !()), and f−1 (A) ∈ . (⇐) We have that is open fuzzy set in (Y; !(s)) if, and only if, −1 ((a; 1]) ∈ s for each a ∈ [0; 1). By the hypothesis, this is equivalent to f−1 ( −1 ((a; 1])) ∈ for each a ∈ [0; 1), i.e. f˜−1 ( ) is open fuzzy set in (X; !()). Corollary 3.2. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ : (X; !()) → (Y; !(s)) be a quotient fuzzy map. If : (a) (X; !()) veri7es the fuzzy version of a property P of topological spaces; (b) the property P is invariant under quotient maps; (c) the fuzzy version of P is a good extension of P. Then (Y; !(s)) veri7es the fuzzy version of P. Corollary 3.3. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ : (X; !()) → (Y; !(s)) be a quotient fuzzy map. If : (a) (Y; !(s)) veri7es the fuzzy version of a property P of topological spaces; (b) the property P is inverse invariant under quotient maps; (c) the fuzzy version of P is a good extension of P. Then (X; !()) veri7es the fuzzy version of P.
545
Proposition 3.4. Let (X; !()); (Y; !(s)) be two weakly induced fuzzy topological spaces; and f˜ be a F-continuous map from (X; !()) onto (Y; !(s)). If there exist a F-continuous map s˜ from (Y; !(s)) to (X; !()) such that f˜ ◦ s˜ = 1Y ; then (Y; !(s)) is F-homeomorphic with X=R; where R is the equivalence relation x1 R x2 ⇔ f(x1 ) = f(x2 ) and the fuzzy topology is the quotient. Proof. We have f˜ ◦ s˜ = 1Y then, by Proposition 2.2 and Remark 2.3, f ◦ s = 1Y . Then, the map g : X=R → Y induced by f is a homeomorphism. Finally, by Propositions 2.4, 2.7 and 2.5, g˜ is onto, one-to-one, F-continuous, and g˜−1 is also F-continuous. Remark 3.5. Using this fuzzy maps, we can generalize to Fuzzy Topology, other results concerning quotient spaces. References & ements de Math&ematique, Topologie G&en&eral, [1] N. Bourbaki, El& Hermann, Paris, 1971. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182 –190. [3] R. Lowen, A comparison of diMerent compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978) 446 – 454. [4] F.G. Lupi&an˜ ez, Fuzzy perfect maps and fuzzy paracompactness, Fuzzy Sets and Systems 98 (1998) 137 –140. [5] F.G. Lupi&an˜ ez, Invariance of fuzzy properties, Mat. Vesnik 50 (1998) 11–16. [6] F.G. Lupi&an˜ ez, Finite-to-one fuzzy maps and fuzzy perfect maps, Kybernetika 34 (1998) 163 –169. [7] P.-M. Pu, Y.-M. Liu, Fuzzy topology I. Neighbourhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571– 599. [8] P.-M. Pu, Y.-M. Liu, Fuzzy topology II. Product and quotient spaces, J. Math. Anal. Appl. 77 (1980) 20 – 37. [9] C.K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45 (1974) 512 – 521. [10] C.K. Wong, Fuzzy topology in fuzzy sets and their applications to cognitive and decision processes, Proc. US –Japan Sem. Univ. California, Berkeley, 1974, Academic Press, New York, 1975, pp. 171–190.