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A note on “On separation axioms in I-fuzzy topological spaces”
Fuzzy Sets and Systems 158 (2007) 1511 – 1513 www.elsevier.com/locate/fss
A note on “On separation axioms in I-fuzzy topological spaces” Fu-Gui Shia,∗ , Hong-Yan Lia, b a Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China b Department of Mathematics, Shandong Institute of Business and Technology, Yantai 264005, China
Received 25 July 2006; received in revised form 27 February 2007; accepted 27 February 2007 Available online 12 March 2007
Abstract As an extension of T0 separation axiom in general topology, RT 0 axiom and s-T0 axiom were introduced in I-fuzzy topological spaces by Yue and Fang [On separation axioms in I-fuzzy topological spaces, Fuzzy Sets and Systems 157 (2006) 780–793]. In this note, it is proved that the RT 0 axiom and the s-T0 axiom are equivalent to each other. © 2007 Elsevier B.V. All rights reserved. Keywords: I-fuzzy topology; RT 0 axiom; s-T0 axiom
The traditional T0 axiom has been generalized to L-topological spaces by many scholars. But as proved in [6], and also noted in [6–8], the fuzzy real line and the fuzzy unit interval need not satisfy the HR-T0 axiom of [1]; and in fact they also need not satisfy Pu and Liu’s T0 axiom of [5]. In 1983, Liu [2] introduced the sub-T0 axiom, for underlying L being a completely distributive DeMorgan algebra, and proved that the fuzzy real line and the fuzzy unit interval satisfy this axiom. Wuyts and Lowen [9] and Rodabaugh [6] independently gave a more general L–T0 axiom, the latter for L being a complete lattice, using only open sets and equivalent to the sub-T0 when L is a completely distributive DeMorgan algebra. Further, Rodabaugh [6] (cf. [7,8]) demonstrated that the L–T0 axiom categorically behaved exactly like the classical T0 axiom in his generalizations of the Isbell and Stone representation theorems and of the Stone–Cech compactification reflector; and Lowen and Srivastava [3,4] proved that the L–T0 axiom categorically behaved exactly like the classical T0 axiom with regard to the epireflective hull of the Sierpinski space in saturated I-topological spaces. In order to extend separation axioms to I-fuzzy topological spaces, Yue and Fang [10] introduced the following definitions: Definition 1. Let (X, ) be an I-fuzzy topological space. (1) Let x be a fuzzy point on X and A ∈ LX . Then x qA ⇔ > A (x), and for U ∈ LX , Qx (U ) =
x ¬qA ⇔ A (x),
(V ).
x qV U
∗ Corresponding author. Tel.: +86 1088583137.
E-mail addresses: [email protected] (F.-G. Shi), [email protected] (H.-Y. Li). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.03.001
1512
F.-G. Shi, H.-Y. Li / Fuzzy Sets and Systems 158 (2007) 1511 – 1513
(2) The degree to which two distinct fuzzy points x and y with x = y are T0 is defined by ⎛ ⎞ ⎛ ⎞ Qy (U )⎠ ∨ ⎝ Qx (V )⎠ . T0 (x , y ) = ⎝ x ¬qU
y ¬qV
(3) The degree to which (X, ) is sub-T0 is defined by ⎧ ⎫ ⎬ ⎨ s-T0 (X, ) = T0 (x , y ) | x = y . ⎩ ⎭ >0
Definition 2. Let (X, ) be an I-fuzzy topological space. (1) The degree to which two distinct crisp points x, y ∈ X with x = y are RT0 is defined as follows: RT0 (x, y) = (U ). U (x)=U (y)
(2) The degree to which (X, ) is RT0 is defined by RT0 (X, ) = {RT0 (x, y) | x = y}. Yue and Fang [10] pointed out that s-T0 (X, ) RT0 (X, ), but its proof was omitted. In this note, we shall prove that the RT0 axiom is equivalent to the s-T0 axiom, i.e., s-T0 (X, ) = RT0 (X, ). Theorem 3. Let (X, ) be an I-fuzzy topological space. Then RT0 (X, ) = s-T0 (X, ). Proof. In order to prove that RT0 (X, ) = s-T0 (X, ), we verify that for any x, y ∈ X, RT0 (x, y) = >0 T0 (x , y ). In fact, for any x, y ∈ X, we have RT0 (x, y) = {(U ) | U (x) = U (y)} = {(U ) | U (x) < U (y)} ∨ {(U ) | U (x) > U (y)} ⎛ ⎞ ⎛ ⎞ =⎝ {(U ) | U (x) > U (y)}⎠ ∨ ⎝ {(U ) | U (y) > U (x)}⎠ >0
⎛ =⎝
⎞
{(U ) | x ¬qU, y qU }⎠ ∨ ⎝
>0
⎛ =⎝
>0 x ¬qU y qV U
=
⎡⎛
⎣⎝
⎞
⎛
(V )⎠ ∨ ⎝ ⎞
⎛
Qy (U )⎠ ∨ ⎝
x ¬qU
>0
=
⎛
>0
⎞
{(U ) | y ¬qU, x qU }⎠
>0
⎞ (V )⎠
>0 y ¬qU x qV U
⎞⎤
Qx (U )⎠⎦
y ¬qU
T0 (x , y )
>0
This shows RT0 (x, y) =
>0 T0 (x , y ). Therefore,
RT0 (X, ) = s-T0 (X, ). The proof is completed.
By Theorem 3 we know that the RT0 axiom and the s-T0 axiom are equivalent to each other.
F.-G. Shi, H.-Y. Li / Fuzzy Sets and Systems 158 (2007) 1511 – 1513
1513
Acknowledgements The authors would like to thank S.E. Rodabaugh and the referees for their valuable comments and suggestions. References [1] B. Hutton, I. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems 3 (1980) 93–104. [2] Y.M. Liu, Pointwise characterizations of complete regularity and embedding theorem in fuzzy topological spaces, Sci. China Ser. A 26 (1983) 138–147. [3] R. Lowen, A.K. Srivastava, Sierpinski objects in subcategories of FTS, Quaestiones Math. 11 (1988) 181–193. [4] R. Lowen, A.K. Srivastava, FTS0 : the epireflective hull of the Sierpinski object in FTS, Fuzzy Sets and Systems 29 (1989) 171–176. [5] P.-M. Pu, Y.-M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [6] S.E. Rodabaugh, A point-set lattice-theoretic framework T for topology which contains LOC as a subcategory of singleton subspaces and in which there are general classes of Stone representation and compactification theorems, First printing February 1986, Second printing April 1987, Youngstown State University Printing Office, Youngstown, OH, USA. [7] S.E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991) 297–345. [8] S.E. Rodabaugh, Categorical frameworks for Stone representation theories, in: S.E. Rodabaugh, E.P. Klement, U. Höhle, (Eds.), Applications of Category Theory to Fuzzy Subsets Theory and Decision Library: Series B: Mathematical and Statistical Methods, vol. 14, Kluwer Academic Publishers, Boston, Dordrecht, London, 1992, pp. 177–231. [9] P. Wuyts, R. Lowen, On local and global measures of separation in fuzzy topological spaces, Fuzzy Sets and Systems 19 (1986) 51–80. [10] Y.L. Yue, J.M. Fang, On separation axioms in I-fuzzy topological spaces, Fuzzy Sets and Systems 157 (2006) 780–793.
A note on “On separation axioms in I-fuzzy topological spaces” Fu-Gui Shia,∗ , Hong-Yan Lia, b a Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China b Department of Mathematics, Shandong Institute of Business and Technology, Yantai 264005, China
Received 25 July 2006; received in revised form 27 February 2007; accepted 27 February 2007 Available online 12 March 2007
Abstract As an extension of T0 separation axiom in general topology, RT 0 axiom and s-T0 axiom were introduced in I-fuzzy topological spaces by Yue and Fang [On separation axioms in I-fuzzy topological spaces, Fuzzy Sets and Systems 157 (2006) 780–793]. In this note, it is proved that the RT 0 axiom and the s-T0 axiom are equivalent to each other. © 2007 Elsevier B.V. All rights reserved. Keywords: I-fuzzy topology; RT 0 axiom; s-T0 axiom
The traditional T0 axiom has been generalized to L-topological spaces by many scholars. But as proved in [6], and also noted in [6–8], the fuzzy real line and the fuzzy unit interval need not satisfy the HR-T0 axiom of [1]; and in fact they also need not satisfy Pu and Liu’s T0 axiom of [5]. In 1983, Liu [2] introduced the sub-T0 axiom, for underlying L being a completely distributive DeMorgan algebra, and proved that the fuzzy real line and the fuzzy unit interval satisfy this axiom. Wuyts and Lowen [9] and Rodabaugh [6] independently gave a more general L–T0 axiom, the latter for L being a complete lattice, using only open sets and equivalent to the sub-T0 when L is a completely distributive DeMorgan algebra. Further, Rodabaugh [6] (cf. [7,8]) demonstrated that the L–T0 axiom categorically behaved exactly like the classical T0 axiom in his generalizations of the Isbell and Stone representation theorems and of the Stone–Cech compactification reflector; and Lowen and Srivastava [3,4] proved that the L–T0 axiom categorically behaved exactly like the classical T0 axiom with regard to the epireflective hull of the Sierpinski space in saturated I-topological spaces. In order to extend separation axioms to I-fuzzy topological spaces, Yue and Fang [10] introduced the following definitions: Definition 1. Let (X, ) be an I-fuzzy topological space. (1) Let x be a fuzzy point on X and A ∈ LX . Then x qA ⇔ > A (x), and for U ∈ LX , Qx (U ) =
x ¬qA ⇔ A (x),
(V ).
x qV U
∗ Corresponding author. Tel.: +86 1088583137.
E-mail addresses: [email protected] (F.-G. Shi), [email protected] (H.-Y. Li). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.03.001
1512
F.-G. Shi, H.-Y. Li / Fuzzy Sets and Systems 158 (2007) 1511 – 1513
(2) The degree to which two distinct fuzzy points x and y with x = y are T0 is defined by ⎛ ⎞ ⎛ ⎞ Qy (U )⎠ ∨ ⎝ Qx (V )⎠ . T0 (x , y ) = ⎝ x ¬qU
y ¬qV
(3) The degree to which (X, ) is sub-T0 is defined by ⎧ ⎫ ⎬ ⎨ s-T0 (X, ) = T0 (x , y ) | x = y . ⎩ ⎭ >0
Definition 2. Let (X, ) be an I-fuzzy topological space. (1) The degree to which two distinct crisp points x, y ∈ X with x = y are RT0 is defined as follows: RT0 (x, y) = (U ). U (x)=U (y)
(2) The degree to which (X, ) is RT0 is defined by RT0 (X, ) = {RT0 (x, y) | x = y}. Yue and Fang [10] pointed out that s-T0 (X, ) RT0 (X, ), but its proof was omitted. In this note, we shall prove that the RT0 axiom is equivalent to the s-T0 axiom, i.e., s-T0 (X, ) = RT0 (X, ). Theorem 3. Let (X, ) be an I-fuzzy topological space. Then RT0 (X, ) = s-T0 (X, ). Proof. In order to prove that RT0 (X, ) = s-T0 (X, ), we verify that for any x, y ∈ X, RT0 (x, y) = >0 T0 (x , y ). In fact, for any x, y ∈ X, we have RT0 (x, y) = {(U ) | U (x) = U (y)} = {(U ) | U (x) < U (y)} ∨ {(U ) | U (x) > U (y)} ⎛ ⎞ ⎛ ⎞ =⎝ {(U ) | U (x) > U (y)}⎠ ∨ ⎝ {(U ) | U (y) > U (x)}⎠ >0
⎛ =⎝
⎞
{(U ) | x ¬qU, y qU }⎠ ∨ ⎝
>0
⎛ =⎝
>0 x ¬qU y qV U
=
⎡⎛
⎣⎝
⎞
⎛
(V )⎠ ∨ ⎝ ⎞
⎛
Qy (U )⎠ ∨ ⎝
x ¬qU
>0
=
⎛
>0
⎞
{(U ) | y ¬qU, x qU }⎠
>0
⎞ (V )⎠
>0 y ¬qU x qV U
⎞⎤
Qx (U )⎠⎦
y ¬qU
T0 (x , y )
>0
This shows RT0 (x, y) =
>0 T0 (x , y ). Therefore,
RT0 (X, ) = s-T0 (X, ). The proof is completed.
By Theorem 3 we know that the RT0 axiom and the s-T0 axiom are equivalent to each other.
F.-G. Shi, H.-Y. Li / Fuzzy Sets and Systems 158 (2007) 1511 – 1513
1513
Acknowledgements The authors would like to thank S.E. Rodabaugh and the referees for their valuable comments and suggestions. References [1] B. Hutton, I. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems 3 (1980) 93–104. [2] Y.M. Liu, Pointwise characterizations of complete regularity and embedding theorem in fuzzy topological spaces, Sci. China Ser. A 26 (1983) 138–147. [3] R. Lowen, A.K. Srivastava, Sierpinski objects in subcategories of FTS, Quaestiones Math. 11 (1988) 181–193. [4] R. Lowen, A.K. Srivastava, FTS0 : the epireflective hull of the Sierpinski object in FTS, Fuzzy Sets and Systems 29 (1989) 171–176. [5] P.-M. Pu, Y.-M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and Moore–Smith convergence, J. Math. Anal. Appl. 76 (1980) 571–599. [6] S.E. Rodabaugh, A point-set lattice-theoretic framework T for topology which contains LOC as a subcategory of singleton subspaces and in which there are general classes of Stone representation and compactification theorems, First printing February 1986, Second printing April 1987, Youngstown State University Printing Office, Youngstown, OH, USA. [7] S.E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991) 297–345. [8] S.E. Rodabaugh, Categorical frameworks for Stone representation theories, in: S.E. Rodabaugh, E.P. Klement, U. Höhle, (Eds.), Applications of Category Theory to Fuzzy Subsets Theory and Decision Library: Series B: Mathematical and Statistical Methods, vol. 14, Kluwer Academic Publishers, Boston, Dordrecht, London, 1992, pp. 177–231. [9] P. Wuyts, R. Lowen, On local and global measures of separation in fuzzy topological spaces, Fuzzy Sets and Systems 19 (1986) 51–80. [10] Y.L. Yue, J.M. Fang, On separation axioms in I-fuzzy topological spaces, Fuzzy Sets and Systems 157 (2006) 780–793.