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On separation axioms in I -fuzzy topological spaces
Fuzzy Sets and Systems 157 (2006) 780 – 793 www.elsevier.com/locate/fss
On separation axioms in I -fuzzy topological spaces夡 Yueli Yue∗ , Jinming Fang Department of Mathematics, Ocean University of China, Qingdao 266071, People’s Republic of China Received 10 September 2003; received in revised form 3 August 2005; accepted 16 November 2005 Available online 5 December 2005
Abstract In this paper, we introduce the notions of some low-level separation axioms and investigate some of their properties and the relations between them in the general framework of I-fuzzy topological spaces. © 2005 Elsevier B.V. All rights reserved. Keywords: I -fuzzy topology; I -fuzzy quasi-coincident neighborhood system; T0 ; T1 ; T2
0. Introduction Since Chang [1] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. In a Chang I-topology, the open sets were fuzzy, but the topology comprising those open sets was a crisp subset of the I-powerset I X . On the other hand, fuzzification of openness was first initiated by Höhle [3] in 1980 and later developed to L-subsets of LX independently by Kubiak [7] and Šostak [16] in 1985. In 1991, from a logical point of view, Ying [17] studied Höhle’s topology and called it fuzzifying topology. As is well known, the neighborhood structure is not suitable to I-topology and Pu and Liu [11] broke through the classical theory of neighborhood system and established the strong and powerful method of quasi-coincident neighborhood system in I-topology. Zhang and Xu [18] established the neighborhood structure in fuzzifying topological spaces. Considering the completeness and usefulness of theory of I-fuzzy topologies, Fang [2] established I-fuzzy quasicoincident neighborhood system in I-fuzzy topological spaces and gave a useful tool to study I-fuzzy topologies. Separation is an essential part of fuzzy topology, on which a lot of work has been done. In the framework of fuzzifying topologies, Shen [15] and Khedr et al. [5] introduced some separation axioms and their separation axioms are discussed on crisp points not on fuzzy points. The aim of this paper is to study some low-level separation axioms on fuzzy points with different support points in I-fuzzy topological spaces. This paper is organized as follows: In Section 1, we give some preliminary concepts and properties. The separation axioms on fuzzy points are introduced by I-fuzzy quasi-coincident neighborhood system in Section 2 and a series of properties of them are investigated. In this section, we also make a comparison between separation axioms defined in this paper and those presented by Rodabaugh [12], Kotzé [6] and Kubiak [8], and give a lot of examples to show the 夡 This
work is supported by Natural Science Foundation of China.
∗ Corresponding author.
E-mail address: [email protected] (Y. Yue). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.11.010
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relations between them. Finally, in Section 3, we do some research on separation axioms in the product spaces and get some important results.
1. Preliminaries In this paper, X is a nonempty set, I = [0, 1] and I0 = (0, 1]. The family of all fuzzy sets on X is denoted by I X and the set of all fuzzy points x (i.e., a fuzzy set A ∈ I X such that A(x) = = 0 and A(y) = 0 for y = x) is denoted by pt (I X ). Let 0X and 1X denote the constant fuzzy set on X taking the value 0 and 1, respectively. Definition 1.1 (Liu and Luo [9], Pu and Liu [11]). Let x ∈ pt (I X ) and A, B ∈ I X . We say x quasi-coincides with A, or say x is quasi-coincident with A, denoted by x qA, if A(x) + > 1. Relation “does not quasi-coincide with’’ or “is not quasi-coincident with’’ is denoted by ¬q. Definition 1.2 (Höhle [3], Ying [17]). A fuzzifying topology on a set X is a map : P (X) → I (where P (X) is the power set of X) such that (1) (X) = (∅) = 1; (2) ∀A, B ∈ P (X), (A ∩ B) (A) ∧ (B); (3) ∀Aj ∈ P (X), j ∈ J , ( j ∈J Uj ) j ∈J (Aj ). The pair (X, ) is called a fuzzifying topological space. The fuzzifying neighborhood system of a point x ∈ X is denoted by Nx , where Nx (A) = x∈B⊆A (B). Definition 1.3 (Höhle and Šostak [4], Kubiak [7], Šostak [16]). An I-fuzzy topology on a set X is a map : I X → I such that (1) (1X ) = (0X ) = 1; (2) ∀U, V ∈ I X , (U ∧ V (V ); ) (U ) ∧ (3) ∀Uj ∈ I X , j ∈ J , ( j ∈J Uj ) j ∈J (Uj ). The pair (X, ) is called an I-fuzzy topological space (I-Ftop, for short). : I X → I is called stratified if, in addition to the above, () = 1 for all ∈ I , where is the constant mapping form X to I and (X, ) is called a stratified I-fuzzy topological space. A map f : (X, ) → (Y, ) is called continuous with respect to I-fuzzy topologies and if (f ← (U )) (U ) for all U ∈ I Y , where f ← is defined by f ← (U )(x) =U (f (x)). A map f : (X, ) → (Y, ) is called open if (U )(f → (U )) for all U ∈ I X , where f → (U )(y) = f (x)=y U (x). f : (X, ) → (Y, ) is called a homeomorphism if and only if f is bijective and f is both continuous and open. Let be an map B : I X → I with B is called a base of if and only if (A) = I-fuzzy topology on X. A X and a map : I X → I is a subbase of if ( ) : I X → I is a base, where B(B ) for all A ∈ I ∈∧ ∈∧ B =A ( ) (A) = ( )∈J B =A ∈J (B ) with ( ) standing for “finite intersection’’. Definition 1.4 (Fang [2]). Let : I X → I be an I-fuzzy topology on X and define Qx : I X → I by Qx (U ) =
x qV U
0
(V )
x qU, x ¬qU.
The set Q = {Qx |x ∈ pt (I X )} is called the I-fuzzy quasi-coincident neighborhood system of . Qx (U ) can be interpreted as the degree to which U is a quasi-coincident neighborhood of x . Lemma 1.5 (Fang [2]). Q = {Qx |x ∈ pt (I X )} defined in Definition 1.4 satisfies: (1) Qx (1X ) = 1 and Qx (0X ) = 0; (2) Qx (U ) > 0 ⇒ x qU ; (3) Qx (U ∧ V ) = Qx (U ) ∧ Qx (V );
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(4) Qx (U ) = y qV Qy (V ); x qV U (5) (U ) = x qU Qx (U ) for all U ∈ I X . Definition 1.6 (Peeters [10], Rodabaugh [14]). Let (X, ) be an I-fuzzy topological space and ∅ = Y ⊆ X. (Y, |Y ) is called the subspace of (X, ), where |Y : I Y → I is defined by (|Y )(U ) = {(V )|V ∈ I X , V |Y = U }. Theorem 1.7. Let f : (X, ) → (Y, ) be a continuous map and ∅ = Z ⊆ X. Then (f |Z) : (Z, |Z) → (Y, ) is continuous, where f |Z : Z → Y is defined by (f |Z)(x) = f (x) for x ∈ Z. Proof. Omitted. Definition 1.8. Let x ∈ pt (I X ) and S be a fuzzy net in I-fuzzy topological space (X, ) . The degrees to which S converges to x and S clusters to x are given as follows, respectively: Con(S, x ) =
{1 − Qx (U )| S is often ¬qU }
and Cl(S, x ) =
{1 − Qx (U )| S is finally ¬qU }.
Definition 1.9. Let (X, ) be an I-fuzzy topological space. If there exists a fuzzifying topology on X such that (A) = r∈I (r (A)) for all A ∈ I X , where r (A) = {x|A(x) > r}, then is called an induced I-fuzzy topology on X, denoted by (), i.e., = (). The pair (X, ) is called the induced I-fuzzy topological space of (X, ). Definition 1.10 (Shen [15]). Let (X, ) be a fuzzifying topological space. The degrees to which (X, ) is T0 , T1 and T2 defined as follows, respectively: ⎧⎛ ⎫ ⎞ ⎬ ⎨ ⎝ T0 (X, ) = Nx (A)⎠ ∨ Ny (B) x, y ∈ X, x = y ; ⎩ ⎭ y ∈A / x ∈B / ⎧⎛ ⎫ ⎞ ⎬ ⎨ ⎝ T1 (X, ) = Nx (A)⎠ ∧ Ny (B) x, y ∈ X, x = y ; ⎩ ⎭ y ∈A / x ∈B / T2 (X, ) = (Nx (A) ∧ Ny (B)) x, y ∈ X, x = y . A∩B=∅
2. T0 , T1 and T2 separation axioms The purpose of this section is to introduce q-T0 , s-T0 , T0 , T1 and T2 separation axioms on fuzzy points in I-fuzzy topological spaces. We will give their properties as well as the relations between them. Definition 2.1. Let (X, ) be an I-fuzzy topological space. (1) The degree to which two distinguished fuzzy points x and x with the same support point are quasi-T0 is defined as follows: ⎛ ⎞ ⎛ ⎞ q-T0 (x , x ) = ⎝ Qx (U )⎠ ∨ ⎝ Qx (V )⎠ . x ¬qU
x ¬qV
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The degree to which (X, ) is quasi-T0 is defined by q-T0 (X, ) = {q-T0 (x , x )|x ∈ X, = }. (2) The degree to which two distinguished fuzzy points x and y with x = y are T0 is defined as follows: ⎛ ⎞ ⎛ ⎞ T0 (x , y ) = ⎝ Qy (U )⎠ ∨ ⎝ Qx (V )⎠ . x ¬qU
y ¬qV
The degree to which (X, ) is T0 is defined by T0 (X, ) = {T0 (x , y )|x , y ∈ pt (I X ), x = y}. The degree to which (X, ) is sub-T0 is defined by ⎧ ⎫ ⎬ ⎨ s-T0 (X, ) = T0 (x , y )|x = y . ⎩ ⎭ ∈I0
(3) The degree to which two distinguished fuzzy points x and y with x = y are T1 is defined as follows: ⎛ ⎞ ⎛ ⎞ T1 (x , y ) = ⎝ Qy (U )⎠ ∧ ⎝ Qx (V )⎠ . x ¬qU
y ¬qV
The degree to which (X, ) is T1 is defined by T1 (X, ) = {T1 (x , y )|x , y ∈ pt (I X ), x = y}. (4) The degree to which two distinguished fuzzy points x and y with x = y are T2 is defined as follows: T2 (x , y ) = (Qy (V ) ∧ Qx (U )). U ∧V =0X
The degree to which (X, ) is T2 is defined by T2 (X, ) = {T2 (x , y )|x , y ∈ pt (I X ), x = y}. If is a Chang I-topology on X , then the concepts in Definition 2.1 coincide with the corresponding ones introduced in [9] if we emphasize the difference between the support points of fuzzy points. The following inequality holds: s-T0 (X, )T0 (X, ) T1 (X, ) T2 (X, ). Firstly, we see two examples. Example 2.2. Let X ⎧ ⎨ 1/4 (A) = 1/5 ⎩ 1
= {x, y} and : I X → I be defined as follows: A(x) > A(y), A(x) < A(y), A(x) = A(y).
Then is an I-fuzzy topology on X. From Definition 2.1 above, we have T0 (x1/2 , y1/2 ) = Furthermore, we can get T0 (x , y ) 41 for all , ∈ I0 . Then T0 (X, ) > T1 (X, ).
1 4
and T1 (x1/2 , y1/2 ) = 15 .
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Example 2.3. Let X be any infinite set and : I X → I be defined as follows: ⎧ A = 0X , 1X , ⎨1 (A) = 21 A satisfying {x|A(x) = 0} is a finite set, ⎩ 0 others. Then is an I-fuzzy topology on X. One can have that T1 (X, ) =
1 2
and T2 (X, ) = 0. Hence T1 (X, ) > T2 (X, ).
Theorem 2.4. Let (X, ) be an I-fuzzy topological space. Then (1) q-T0 (x , x ) = Qx ((x ) ) ( < ); (2) T0 (x , y ) = Qy ((x ) ) ∨ Qx ((y ) ); (3) T1 (x , y ) = Qy ((x ) ) ∧ Qx ((y ) ). Proof. From the definition of Qx , it is easy to prove it and omitted.
Theorem 2.5. Let (Y, |Y ) be the subspace of (X, ). (1) For any two distinguished fuzzy points x and x in (Y, |Y ), we have q-T0Y (x , x ) = q-T0X (x , x ). Hence q-T0 (X, )q-T0 (Y, |Y ). (2) For any two distinguished fuzzy points x and y with x = y in (Y, |Y ), we have T0Y (x , y ) = T0X (x , y ). Hence s-T0 (X, )s-T0 (Y, |Y ) and T0 (X, ) T0 (Y, |Y ). (3) For any two distinguished fuzzy points x and y with x = y in (Y, |Y ), we have T1Y (x , y ) = T1X (x , y ). Hence T1 (X, )T1 (Y, |Y ). (4) T2 (X, )T2 (Y, |Y ). Proof. We only prove (2) and the others can be obtained in a similar way and omitted. For any two distinguished fuzzy points x and y with x = y in I Y , from Theorem 2.4, we have T0Y (x , y ) = QYy ((x ) ) ∨ QYx ((y ) ), and X T0X (x , y ) = QX y ((x ) ) ∨ Qx ((y ) ). Y In order to prove that T0Y (x , y ) = T0X (x , y ), it suffices to show that QYx ((y ) ) = QX x ((y ) ) and Qy ((x ) ) = X Qy ((x ) ). In fact,
QYx ((y ) ) =
x qW (y )
|Y (W ) =
x qW (y ) V |Y =W
(V ) =
x qU (y )
(U ) = QX x ((y ) ).
Similarly, we can get QYy ((x ) ) = QX y ((x ) ), as desired. Naturally, we have s-T0 (X, ) s − T0 (Y, |Y ) and T0 (X, ) T0 (Y, |Y ).
Example 2.6. Let X = {x, y, z} and : I X → I be defined as follows: ⎧ 9/10 A(x) = A(y) > A(z), ⎪ ⎪ ⎪ ⎪ 8/10 A(x) = A(z) > A(y), ⎪ ⎪ ⎪ ⎪ ⎨ 7/10 A(y) = A(z) > A(x), (A) = 8/10 A(x) > A(y) and A(x) > A(z), ⎪ ⎪ 7/10 A(z) > A(x) and A(z) > A(y), ⎪ ⎪ ⎪ ⎪ 7/10 A(y) > A(x) and A(y) > A(z), ⎪ ⎪ ⎩ 1 others.
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Then is an I-fuzzy topology on X. Let Y = {y, z} ⊆ X. Then we can obtain that ⎧ ⎨ 9/10 B(y) > B(z), |Y (B) = 8/10 B(y) < B(z), ⎩ 1 B(y) = B(z). It is easy to verify that T1 (X, ) = 7/10 < 8/10 = T1 (Y, |Y ). Example 2.7. In Example 2.3, let Y = {x, y} ⊆ X. We can get |Y (B) = 1 for all B ∈ I Y . Hence T2 (Y, |Y ) = 1 > 0 = T2 (X, ). Theorem 2.8. Let (X, ) be an I-fuzzy topological space. Then T1 (X, ) = {((x ) )|x ∈ X, ∈ I0 }. Proof. For any two fuzzy points x , y with x = y, from (5) of Lemma 1.5, we have ⎫ ⎧ ⎬ ⎨ {((z ) )|z ∈ X, ∈ I0 } = Qw ((z ) )|z ∈ X, ∈ I0 ⎭ ⎩ w q(z ) Qw ((x ) ) w q(x )
Qy ((x ) ).
Similarly, we have {((z ) )|z ∈ X, ∈ I0 } Qx ((y ) ). Then {((z ) )|z ∈ X, ∈ I0 } T1 (x , y ). Hence
{((z ) )|z ∈ X, ∈ I0 } T1 (X, ).
On the other hand, we have T1 (X, ) = {Qy ((x ) ) ∧ Qx ((y V ) )|, ∈ I0 , x = y} {Qy ((x ) )|, ∈ I0 , x = y} ⎧ ⎫ ⎬ ⎨ = Qy ((x ) )| ∈ I0 , x ∈ X ⎩ ⎭ y ∈pt (I X ),y=x ⎧ ⎫ ⎬ ⎨ Qy ((x ) )| ∈ I0 , x ∈ X ⎩ ⎭ y q(x ) = {((x ) )| ∈ I0 , x ∈ X}. So T1 (X, ) = {((x ) )|x ∈ X, ∈ I0 }. Remark 2.9. In [6], Kotzé introduced T1 in (L, M)-topological spaces. An (L, M)-topological spaces (X, L, M, ) is T1 if and only if the map : LX → M assigns to the complement of a fuzzy point on X the value 1 in M. The following example will show that T1 defined in this paper is different from Kotzé’s. Example 2.10. Let X ⎧ ⎨ 9/10 (A) = 8/10 ⎩ 1
= {a, b}, L = M = I and be defined by A(a) > A(b), A(a) < A(b), A(a) = A(b).
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Then it is easy to verify that is an I-fuzzy topology on X. We can get T1 (X, ) = 8/10 according to our definition of T1 . Since ((a1/2 ) ) = 8/10 = 1, we know that (X, I, I, ) is not T1 in W. Kotzé’s sense. From this example, we know that what we give in this paper is the degree to which one I-fuzzy topological space is T1 and the value of the degree is in [0,1], which is very different from Kotzé’s. Question 2.11. In paper [6], Kotzé also introduced sober and semi-sober in (L, M)-topological spaces. How to introduce the corresponding sober and semi-sober in I-fuzzy topological spaces and discuss the relations with our T0 , T1 and T2 ? In [12,13], Rodabaugh defined a standard T0 axiom (denoted by RT0 in this paper) in L-topological spaces as follows: RT0 axiom: ∀x, y ∈ X such that x = y, there exists an open set u such that u(x) = u(y). In [8], Kubiak defined T1 and T2 axioms (denoted by KT1 and KT2 , respectively, in this paper) in L-topological spaces as follows: (1) KT1 axiom: If x = y, there exist two open set u, v such that u(x)u(y) and v(y)v(x). (2) KT2 axiom: If x = y, there exist two open set u, v such that u(x)u(y), v(y)v(x) and uv . If we replace the value lattice I = [0, 1] with {0, 1} in Definition 1.3, i.e., : I X → {0, 1} (or is a Chang I-topology), the readers can easily verify that the definitions of T0 , T1 and T2 defined in this paper imply RT0 , KT1 and KT2 , respectively. Besides this, we have more generalized result (See Theorem 2.12). Let (X, ) be an I-fuzzy topological space. We can rewritten Rodabaugh’s and Kubiak’s separation axioms for the I-Ftop setting as follows: (1) The degree to which two distinguished crisp points x, y ∈ X with x = y are RT0 is defined as follows: RT0 (x, y) = (u). u(x)=u(y)
The degree to which (X, ) is RT0 is defined by RT0 (X, ) = {RT0 (x, y)|x = y}. (2) The degree to which two distinguished crisp points x, y ∈ X with x = y are KT1 is defined as follows: KT1 (x, y) = (u) ∧ (v) = (u) ∧ (v). u(x)u(y)
v(y)u(x)
u(x)>u(y)
v(y)>v(x)
The degree to which (X, ) is KT1 is defined by KT1 (X, ) = {KT1 (x, y)|x = y}. (3) The degree to which two distinguished crisp points x, y ∈ X with x = y are KT2 is defined as follows: KT2 (x, y) = (u) ∧ (v) = (u) ∧ (v), u(x)u(y), v(y)v(x), u v
(u,v)∈D (x,y)
where D(x, y) = {(u, v)|u(x) > u(y), v(y) > v(x), uv }. The degree to which (X, ) is KT2 is defined by KT2 (X, ) = {KT2 (x, y)|x = y}. About the relation between above generalizations and ours, we have the following theorem. Theorem 2.12. Let (X, ) be an I-fuzzy topological spaces. The following formulas are valid: (1) s-T0 (X, )RT0 (X, ), (2) T1 (X, ) KT1 (X, ), (3) T2 (X, ) KT2 (X, ).
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Proof. We only prove (2) and (3). (2) Let t < T1 (X, ). On account of t < T1 (X, )T1 (x1 , y1 ) = Qy1 ((x1 ) ) ∧ Qx1 ((y1 ) ) =
y1 qB (x1 )
(B) ∧
787
(A)
x1 qA (y1 )
for all x = y, there exists A ∈ I X such that x1 qA (y1 ) and t (A), and there exists B ∈ I X such that y1 qB (x1 ) and t (B), respectively. From x1 qA (y1 ) and y1 qB (x1 ) , we have A(x) > 0 = A(y) and B(y) > 0 = B(x). Hence t u(x)>u(y) (u) ∧ v(y)>v(x) (v). Therefore, t KT1 (X, ). From the arbitrariness of t, we have T1 (X, )KT1 (X, ), as desired. (3) Let t < T2 (X, ). Then t < A∧B=0X Qy1 (B) ∧ Qx1 (A) for all x = y . Hence there exist A, B ∈ I X such that A ∧ B = 0X and t < Qy1 (B) ∧ Qx1 (A). Furthermore, there exists C ∈ I X such that x1 qC A and t (C). Similarly, there exists D ∈ I X such that y1 qD B and t (D). By x1 qC A, y1 qD B and A ∧ B = 0X , we have C(x) > C(y), D(y) > D(x) and C D . Hence t (C)∧(D) (u,v)∈D(x,y) (u)∧(v). Therefore, t KT2 (X, ). Thus the conclusion. Remark 2.13. One thing we want to point out that the separation axioms of Rodabaugh’s and Kubiak’s are defined on crisp points and what we do in this paper is on fuzzy points. Besides the “separation degree’’ of points in background, we must also consider the same problem for points in different layers at the same time. This can be also seen from the proof of Theorem 2.12. When we prove Theorem 2.12, we obtain the results only using the fuzzy points with height 1. Now we see another example. Example 2.14. We consider the I-fuzzy unit interval I (I ) in I-topological spaces. For details about I (I ), please refer to [8,9]. It can be also regarded as I-fuzzy topology according to the characteristic function, i.e., 1 A ∈ I (I ), I (I )(A) = 0 A∈ / I (I ). We will show that q-T0 (X, I (I )) = T0 (X, I (I )) = 0. Hence T1 (X, I (I )) = T1 (X, I (I )) = 0. Firstly, we have q-T0 (X, I (I )) = 0. Let x ∈ X be defined as follows: ⎧ t < 0, ⎨ 1 x(t) = 1/2 0 t 1, ⎩ 0 t > 1. Then q-T0 (X, I (I ))q-T0 (x2/3 , x3/4 ) = Qx3/4 ((x2/3 ) ) = x3/4 qA (x2/3 ) I (I )(A) = 0. Secondly, we have T0 (X, I (I )) = 0. Let x, y ∈ X be defined as follows: ⎧ t < 0, ⎨ 1 x(t) = 1/3 0 t 1, ⎩ 0 t > 1. ⎧ t < 0, ⎨ 1 y(t) = 1/2 0 t 1, ⎩ 0 t > 1. Then T0 (X, I (I )) T0 (x1/3 , y1/3 ) = Qy1/3 ((x1/3 ) ) ∨ Qx1/3 ((y1/3 ) ) = I (I )(U ) ∨ y1/3 qU (x1/3 )
= 0.
x1/3 qV (y1/3 )
I (I )(V )
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Finally, we have T1 (X, I (I )) = T2 (X, I (I )) = 0. It is easy to verify that s-T0 (X, I (I )) = RT0 (X, I (I )) = KT1 (X, I (I )) = KT2 (X, I (I )) = 1. This is to say the degree to which I (I ) is KT2 (resp. KT1 , RT0 and s-T0 ) is 1 or I (I ) is KT2 (resp. KT1 , RT0 and s-T0 ). Remark 2.15. From the above example, we know that the degrees to which I (I ) is q-T0 , T0 , T1 and T2 are 0. So the I-fuzzy unit interval is not compatible with the T0 , T1 and T2 separation axioms proposed in this paper. It is an open to construct a model of R for I-Ftop and construct over separation axioms for I-Ftop so that this model is compatible (at least to some degree) with these axioms. Theorem 2.16. If (X, ) is an induced I-fuzzy topological space, i.e., there exists one fuzzifying topology on X such that = (), then (1) s-T0 (X, ) = T0 (X, ) = T0 (X, ), (2) T1 (X, ) = T1 (X, ), (3) T2 (X, ) = T2 (X, ). Proof. We prove (1) and (3), the proof of (2) is similar to that of (1). (1) At first, we want to prove T0 (X, ) T0 (X, ), this is to say T0 (X, ) Qy ((x ) ) ∨ Qx ((y ) ) for any two fuzzy points x , y with x = y. Let x , y ∈ pt (I X ) such that x = y and < T0 (X, ). On account of x = y, we have < Nx (X − {y}) or < Ny (X − {x}). No loss of generality, suppose < Nx (X − {y}), then there exists C ⊆ X such that x ∈ C ⊆ X − {y} and (C). Since (X, ) is induced I-fuzzy topological space, we have (C) = ( C ) Qx ( C )Qx ((y ) ), where C is the characteristic function of C. Hence Qy ((x ) ) ∨ Qx ((y ) ). Thus T0 (X, ). Therefore, T0 (X, ) T0 (X, ). On the other hand, we need to prove that s-T0 (X, ) Nx (X − {y}) ∨ Ny (X − {x}) for any x, y ∈ X with x = y. Let < s-T0 (X, ). Then for any x, y ∈ X with x = y, there exists ∈ I0 such that < T0 (x , y ). Hence < Qx ((y ) ) X or < Qy ((x ) ). For convenience, we assume that < Qx ((y ) ). Then there exists H ∈ I such that x qH (y ) and < (H ) = r∈I (r (H )). Let A = 1− (H ). Then we have x ∈ A, y ∈ / A and < (A) Nx (X − {y}). Thus, T0 (X, ). So s-T0 (X, ) T0 (X, ). (3) Let x, y be any two points in X such that x = y and let < T2 (X, ). Then there exist U, V ∈ I X with X U ∧ V = 0X such that < Qx1 (U ) and < Qy1 (V ). Then there exist W, H ∈ I with x1 qW U and y1 qH V such that < (W ) = r∈I (r (W )) and < (H ) = r∈I (r (H )). Now let A = 0 (W ) and B = 0 (H ). Then x ∈ A, y ∈ B, A∩B = ∅, < (A) qNx (A) and < (B) Ny (B). Thus T2 (X, ). Hence T2 (X, ) T2 (X, ). Conversely, let x , y be any two fuzzy points with x = y and let < T2 (X, ). Since x = y, there exist A, B ⊆ X with A ∩ B = ∅ such that < Nx (A) and < Ny (B). By < Nx (A), there exists C ⊆ X such that x ∈ C ⊆ A and < (C). By < Ny (B), there exists D ⊆ X such that y ∈ D ⊆ B and < (D). It is easy to check that x q C , y q D and C ∧ D = 0X . Since (X, ) is induced I-fuzzy topological space, we have (C) = ( C ) Qx ( C ) and (D) = ( D )Qy ( D ). Hence Qx ( C ) ∧ Qy ( D ). Therefore, T2 (X, ), as desired. Theorem 2.17. If f : (X, ) → (Y, ) is a homeomorphism. Then (1) q-T0 (x , x ) = q-T0 (f → (x ), f → (x )); (2) T0 (x , y ) = T0 (f → (x ), f → (y )); (3) T1 (x , y ) = T1 (f → (x ), f → (y )); (4) T2 (x , y ) = T2 (f → (x ), f → (y )). Proof. Omitted. Theorem 2.18. Let(X, ) be an I-fuzzy topological space. Then T2 (X, ) = {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in I X }. Proof. Let T2 (X, ) < . Then there exist two fuzzy points x , y with x = y such that Qx (V ) < or Qy (U ) < when U ∧ V = 0X . Let (Qx ) = {A|Qx (A) } and (Qy ) = {B|Qy (B) }. It is easy to prove that (Qx ) ×
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789
(Qy ) is a co-directed set. Furthermore, if A ∈ (Qx ) and B ∈ (Qy ) , then A ∧ B = 0X . Hence there exists x(A,B) ∈ X such that (A ∧ B)(x(A,B) ) > 0. Define S ∗ : (Qx ) × (Qy ) → pt (I X ) by S ∗ (A, B) = (x(A,B) )1−1/2[(A∧B)(x(A,B) )] for (A, B) ∈ (Qx ) × (Qy ) , then S ∗ is a fuzzy net. Now we want to show 1 − Con(S ∗ , x ) ∧ Con(S ∗ , y ) , i.e., {Qx (A)|S ∗ is often¬qA} ∨ {Qy (B)|S ∗ is often¬qB} . Suppose A ∈ I X such that S ∗ is often ¬qA. If Qx (A) , then A ∈ (Qx ) . Take B ∈ (Qy ) , then when (E, F ) ∈ (Qx ) × (Qy ) and (E, F ) ⊆ (A, B), we have, A(x(E,F ) ) + S ∗ ((E, F ))(x(E,F ) ) = A(x(E,F ) ) + 1 − 1/2(E ∧ F )(x(E,F ) ) > A(x(E,F ) ) + 1 − (E ∧ F )(x(E,F ) ) A(x(E,F ) ) + 1 − (A ∧ B)(x(E,F ) ) 1. Hence S(E, F )qA, i.e., S ∗ is eventually qA. This is contradict to the supposition. Therefore, Qx (A) < . Hence {Qx (A)| S is often ¬qA} . Similarly, {Qy (B)|Sis often¬qB} , as desired. Then {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . From the arbitrariness of , we have T2 (X, ) {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X}. Conversely, if T2 (X, ) > , then there exists U, V ∈ I X with U ∧ V = 0X such that Qx (U ) > and Qy (V ) > for any two fuzzy points x , y with x = y. For any fuzzy net S, if Con(S, x ) 1 − , then Qx (U ) when S is often ¬qU . Now we assert that {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . If not, then there exist a fuzzy net S and x , y with x = y such that 1 − Con(S, x ) ∧ Con(S, y ) , i.e., Con(S, x )1− and Con(S, y ) 1− . From Con(S, x ) 1− and Qx (U ) > , we know that S is eventually qU . Similarly, S is eventually qV . Hence S is eventually q(U ∧ V ). So U ∧ V = 0X , this is a contradiction. Therefore, {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . Hence T2 (X, )
{1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X}.
Thus the conclusion.
3. Separation axioms in the product space In this section,we study separation axioms in the product space. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces and P : j ∈J Xj → X be the projection. Then the product I-fuzzy topology whose subbase is defined by
∀W ∈ I j ∈J
Xj
, (W ) =
j ∈J
Pj← (U )=W
j (U ),
790
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and this product topology is denoted by j ∈J j . We know that if X (y) is a stratified factor space of ( ∈J X , ∈J ) parallel to X through y = (y ) ∈J , then (P |X (y))→ : (X (y), ∈J |X (y)) → (X , ) is a homeomorphism. Theorem 3.1. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces. Then (1) j ∈J q-T0 (Xj , j ) q-T0 ( j ∈J Xj , j ∈J j ); (2) j ∈J T0 (Xj , j )T0 ( j ∈J Xj , j ∈J j ); (3) j ∈J T1 (Xj , j )T1 ( j ∈J Xj , j ∈J j ); (4) j ∈J T2 (Xj , j )T2 ( j ∈J Xj , j ∈J j ). In addition, if (Xj , j ) is stratified for all j ∈ J , then the following equalities are true: (5) j ∈J q-T0 (Xj , j ) = q-T0 ( j ∈J Xj , j ∈J j ); (6) j ∈J T0 (Xj , j ) = T0 ( j ∈J Xj , j ∈J j ); (7) j ∈J T1 (Xj , j ) = T1 ( j ∈J Xj , j ∈J j ); (8) j ∈J T2 (Xj , j ) = T2 ( j ∈J Xj , j ∈J j ). Proof. We prove (3) and(4). (3) Let x , y ∈ pt (I j ∈J Xj ) be any two fuzzy points with x = y and let r ∈ I0 be any number with the property r< T1 (Xj , j ). Clearly, r < T1 (Xj , j ) for each j ∈ J . Due to x = y, there exists k ∈ J such that Pk (x) = Pk (y). j ∈J
We denote Pk (x), Pk (y) by a and b, respectively. Hence a , b ∈ pt (I Xk ) with a = b. Therefore, r < Qb ((a ) ) ∧ Qa ((b ) ). Then there exists C, D ∈ I Xk with b qC (a ) and a qD (b ) such that r < k (C) and r < k (D). Thus x qPk← (D)(y ) and y qPk← (C) (x ) . So j (Pk← (C))Qy (Pk← (C))Qy ((x ) ), r < k (C) j ∈J
and r < k (D)
j (Pk← (D))Qy (Pk← (D))Qx ((y ) ).
j ∈J
This means that
⎛ ⎞ Xj , Xj ⎠ . r {T1 (x , y )|x , y ∈ pt (I j ∈J Xj ), x = y} = T1 ⎝ j ∈J
j ∈J
From the arbitrariness of r, we get that ⎛ ⎞ T1 (Xj , j )T1 ⎝ Xj , j ⎠ . j ∈J
(4) Let hs , gt ∈ pt (I
j ∈J
j ∈J
Xj
) be any two fuzzy points with h = g and let r < ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ∀j ∈ J, r < Qxs (Aj ) ∧ Qyt (Bj )|s, t ∈ I0 , x = y . ⎪ ⎪ ⎩Aj ∧Bj =0X ⎭ j ∈J
j
j ∈J
T2 (Xj , j ). Then we have,
Y. Yue, J. Fang / Fuzzy Sets and Systems 157 (2006) 780 – 793
791
Since h = g, there exists j ∈ J such that h(j ) = g(j ). And then there exist G, H ∈ I Xj with G ∧ H = 0Xj such that h(j )s qG, g(j )t qH , j (G) > r and j (H ) > r. On account of Pj← (G) ∧ Pj← (H ) = 0 Xj , we can get ⎛
Qhs (Pj← (G)) ⎝
j ∈J
⎞ j ⎠ (Pj← (G))j (G) > r.
j ∈J
Similarly, Qgt (Pj← (G)) > r. Hence (Qhs (A) ∧ Qgt (B))Qhs (Pj← (G)) ∧ Qgt (Pj← (G)) > r. A∧B=0j ∈J
Xj
Therefore,
⎛
T2 (Xj , j )T2 ⎝
j ∈J
Xj ,
j ∈J
⎞ j ⎠ .
j ∈J
In addition, assume that (X j , j ) isstratified for j ∈ J , then (Xj , j ) is homeomorphic to some subspace (Xj (y), j ∈J j |Xj (y)) of ( j ∈J Xj , j ∈J j ). From Theorems 2.5 and 2.17, we have ⎛ ⎞ Xj , j ⎠ T2 (Xj (y), j |Xj (y)) = T2 (Xj , j ). T2 ⎝ j ∈J
j ∈J
j ∈J
j ∈J
j ∈J
Similarly, we have ⎛ ⎞ Xj , j ⎠ T1 (Xj (y), j |Xj (y)) = T1 (Xj , j ), T1 ⎝ j ∈J
⎛ T0 ⎝
j ∈J
Xj ,
j ∈J
and
⎛ q-T0 ⎝
j ∈J
⎞ j ⎠
j ∈J
Xj ,
j ∈J
Thus the conclusion.
j ∈J
T0 (Xj (y),
j ∈J
j ∈J
⎞ j ⎠
j ∈J
j |Xj (y)) =
j ∈J
q-T0 (Xj (y),
j ∈J
T0 (Xj , j )
j ∈J
j |Xj (y)) =
j ∈J
q-T0 (Xj , j ).
j ∈J
Example 3.2. Let X = {x} and Y = {y}. Define 1 : I X → I by 1 A ∈ {0X , 1X }, 1 (A) = 0 others. and define 2 : I Y → I by 2 (A) = 1 for all A ∈ I Y , respectively. Then 1 is an I-fuzzy topology on X and 2 is an I-fuzzy topology on Y. We can also calculate that the product topology 1 × 2 of 1 and 2 satisfies 1 × 2 (A) = 1 for all A ∈ I X×Y . It is easy to verify that q-T0 (X, 1 ) = 0, q-T0 (Y, 2 ) = 1 and q-T0 (X × Y, 1 × 2 ) = 1. Hence q-T0 (X, 1 ) ∧ q-T0 (Y, 2 ) < q-T0 (X × Y, 1 × 2 ). Theorem 3.3. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces. Then ⎛ ⎞ s-T0 (Xj , j ) = s-T0 ⎝ Xj , j ⎠ . j ∈J
j ∈J
j ∈J
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Y. Yue, J. Fang / Fuzzy Sets and Systems 157 (2006) 780 – 793
Proof. Let r < S-T0 ( j ∈J Xj , j ∈J j ) and j0 ∈ J . For any two distinguished points a, b ∈ Xj0 , take x = (xj )j ∈J ∈ j ∈J Xj and y = (yj )j ∈J ∈ j ∈J Xj such that xj0 = a, yj0 = b and xj = yj whenever j ∈ J − {j0 }. Then there exists ∈ I0 such that r < T0 (x , y ). Hence ( ) (Al ) r < Qy ((x ) ) = y qG (x )
or r < Qx ((y ) ) =
l∈L
Al =G l∈L
x qH (y )
m∈M
( ) (Bm ).
Am =H m∈M
For convenience, we assume that r < Qy ((x ) ). Then there exist Gx (x ) and a family {Axl }l∈L such that y qGx = ( ) x x Al and r < (Al ). Furthermore, there exists l ∈ L such that y qAxl and r < ( ) (Axl ). On account of
l∈L
l∈L
( ) (Axl ) =
j (S),
( )n∈Ll Cn =Axl n∈Ll j ∈J Pj← (S)=Cn
there exists a finite family of {Cn }n∈Ll such that ( )n∈Ll Cn = Axl , and ∀n ∈ Ll , there exist Sjn ∈ I Xj such that (Sjn ) P ← (Sjn ) = Cn and jn (Sjn ) > r; Considering x ¬qAxl and y qAxl , then there exists n ∈ Ll such that x ¬qPj← n ← (S )(x) = P ← (S )(y). By the definition of x and y, we can assert that j = j . and y qPj← (S ). Therefore, P j j j n 0 n n n jn jn n Hence r < jn (Sjn )Qb (Sjn ) Qb ((a ) ). Thus, s-T0 (Xj0 , j0 ) =
⎧ ⎨ ⎩
T0 (x , y )|x = y
∈I0
⎫ ⎬ ⎭
r.
Therefore,
⎛ s-T0 (Xj , j ) s-T0 ⎝
j ∈J
The proof of
j ∈J
j ∈J
Xj ,
⎞ j ⎠ .
j ∈J
s-T0 (Xj , j ) s-T0 (
j ∈J
Xj ,
j ∈J
j ) is similar to that of (3) in Theorem 3.1.
Acknowledgements The authors would like to thank Prof. S.E. Rodabaugh and the anonymous referees for their very helpful comments and valuable suggestions.
References [1] [2] [3] [4]
C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–193. J. Fang, I -FTOP is isomorphic to I -FQN and I -AITOP, Fuzzy Sets and Systems 147 (2004) 317–325. U. Höhle, Uppersemicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980) 659–673. U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 123–272, (Chapter 3). [5] F.H. Khedr, F.M. Zeyada, O.R. Sayed, On separation axioms in fuzzifying topology, Fuzzy Sets and Systems 119 (2001) 439–458.
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[6] W. Kotzé, Lifting of sobriety concepts with particular reference to (L, M)-topological spaces, in: S.E. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets, A Handbook of Recent Developments in the Mathematics of Fuzzy Sets Series: Trends in Logic, vol. 20, Kluwer Academic Publishers, 2003, pp. 415–426, (Chapter 16). [7] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985. [8] T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73 (1995) 25–53. [9] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1997. [10] W. Peeters, Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures, Fuzzy Sets and Systems 104 (1999) 423–433. [11] B.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. [12] S.E. Rodabaugh, A point set lattice-theoretic framework T which contains LOC as a subcategory of singleton spaces and in which there are general classes of Stone representation and compactification theorems, first draft February 1986/ second draft April 1987, Youngstown State University Central Printing Office, Youngstown, Ohio, 1987. [13] S.E. Rodabaugh, Applications of localic separation axioms compactness axioms representations, and compactifications to poslat topological spaces, Fuzzy Sets and Systems 73 (1995) 55–87. [14] S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 273–388, (Chapter 4). [15] J. Shen, Separation axiom in fuzzifying topology, Fuzzy Sets and Systems 57 (1993) 227–238. [16] A.P. Šostak, on a fuzzy topological structure, Rend. Circ. Mat. Palermo (Suppl. Ser. II) 11 (1985) 89–103. [17] M. Ying, A new approach to fuzzy topology (I), Fuzzy Sets and Systems 39 (1991) 303–321. [18] D. Zhang, L. Xu, Categories isomorphic to FNS, Fuzzy Sets and Systems 104 (1999) 373–380.
On separation axioms in I -fuzzy topological spaces夡 Yueli Yue∗ , Jinming Fang Department of Mathematics, Ocean University of China, Qingdao 266071, People’s Republic of China Received 10 September 2003; received in revised form 3 August 2005; accepted 16 November 2005 Available online 5 December 2005
Abstract In this paper, we introduce the notions of some low-level separation axioms and investigate some of their properties and the relations between them in the general framework of I-fuzzy topological spaces. © 2005 Elsevier B.V. All rights reserved. Keywords: I -fuzzy topology; I -fuzzy quasi-coincident neighborhood system; T0 ; T1 ; T2
0. Introduction Since Chang [1] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. In a Chang I-topology, the open sets were fuzzy, but the topology comprising those open sets was a crisp subset of the I-powerset I X . On the other hand, fuzzification of openness was first initiated by Höhle [3] in 1980 and later developed to L-subsets of LX independently by Kubiak [7] and Šostak [16] in 1985. In 1991, from a logical point of view, Ying [17] studied Höhle’s topology and called it fuzzifying topology. As is well known, the neighborhood structure is not suitable to I-topology and Pu and Liu [11] broke through the classical theory of neighborhood system and established the strong and powerful method of quasi-coincident neighborhood system in I-topology. Zhang and Xu [18] established the neighborhood structure in fuzzifying topological spaces. Considering the completeness and usefulness of theory of I-fuzzy topologies, Fang [2] established I-fuzzy quasicoincident neighborhood system in I-fuzzy topological spaces and gave a useful tool to study I-fuzzy topologies. Separation is an essential part of fuzzy topology, on which a lot of work has been done. In the framework of fuzzifying topologies, Shen [15] and Khedr et al. [5] introduced some separation axioms and their separation axioms are discussed on crisp points not on fuzzy points. The aim of this paper is to study some low-level separation axioms on fuzzy points with different support points in I-fuzzy topological spaces. This paper is organized as follows: In Section 1, we give some preliminary concepts and properties. The separation axioms on fuzzy points are introduced by I-fuzzy quasi-coincident neighborhood system in Section 2 and a series of properties of them are investigated. In this section, we also make a comparison between separation axioms defined in this paper and those presented by Rodabaugh [12], Kotzé [6] and Kubiak [8], and give a lot of examples to show the 夡 This
work is supported by Natural Science Foundation of China.
∗ Corresponding author.
E-mail address: [email protected] (Y. Yue). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.11.010
Y. Yue, J. Fang / Fuzzy Sets and Systems 157 (2006) 780 – 793
781
relations between them. Finally, in Section 3, we do some research on separation axioms in the product spaces and get some important results.
1. Preliminaries In this paper, X is a nonempty set, I = [0, 1] and I0 = (0, 1]. The family of all fuzzy sets on X is denoted by I X and the set of all fuzzy points x (i.e., a fuzzy set A ∈ I X such that A(x) = = 0 and A(y) = 0 for y = x) is denoted by pt (I X ). Let 0X and 1X denote the constant fuzzy set on X taking the value 0 and 1, respectively. Definition 1.1 (Liu and Luo [9], Pu and Liu [11]). Let x ∈ pt (I X ) and A, B ∈ I X . We say x quasi-coincides with A, or say x is quasi-coincident with A, denoted by x qA, if A(x) + > 1. Relation “does not quasi-coincide with’’ or “is not quasi-coincident with’’ is denoted by ¬q. Definition 1.2 (Höhle [3], Ying [17]). A fuzzifying topology on a set X is a map : P (X) → I (where P (X) is the power set of X) such that (1) (X) = (∅) = 1; (2) ∀A, B ∈ P (X), (A ∩ B) (A) ∧ (B); (3) ∀Aj ∈ P (X), j ∈ J , ( j ∈J Uj ) j ∈J (Aj ). The pair (X, ) is called a fuzzifying topological space. The fuzzifying neighborhood system of a point x ∈ X is denoted by Nx , where Nx (A) = x∈B⊆A (B). Definition 1.3 (Höhle and Šostak [4], Kubiak [7], Šostak [16]). An I-fuzzy topology on a set X is a map : I X → I such that (1) (1X ) = (0X ) = 1; (2) ∀U, V ∈ I X , (U ∧ V (V ); ) (U ) ∧ (3) ∀Uj ∈ I X , j ∈ J , ( j ∈J Uj ) j ∈J (Uj ). The pair (X, ) is called an I-fuzzy topological space (I-Ftop, for short). : I X → I is called stratified if, in addition to the above, () = 1 for all ∈ I , where is the constant mapping form X to I and (X, ) is called a stratified I-fuzzy topological space. A map f : (X, ) → (Y, ) is called continuous with respect to I-fuzzy topologies and if (f ← (U )) (U ) for all U ∈ I Y , where f ← is defined by f ← (U )(x) =U (f (x)). A map f : (X, ) → (Y, ) is called open if (U )(f → (U )) for all U ∈ I X , where f → (U )(y) = f (x)=y U (x). f : (X, ) → (Y, ) is called a homeomorphism if and only if f is bijective and f is both continuous and open. Let be an map B : I X → I with B is called a base of if and only if (A) = I-fuzzy topology on X. A X and a map : I X → I is a subbase of if ( ) : I X → I is a base, where B(B ) for all A ∈ I ∈∧ ∈∧ B =A ( ) (A) = ( )∈J B =A ∈J (B ) with ( ) standing for “finite intersection’’. Definition 1.4 (Fang [2]). Let : I X → I be an I-fuzzy topology on X and define Qx : I X → I by Qx (U ) =
x qV U
0
(V )
x qU, x ¬qU.
The set Q = {Qx |x ∈ pt (I X )} is called the I-fuzzy quasi-coincident neighborhood system of . Qx (U ) can be interpreted as the degree to which U is a quasi-coincident neighborhood of x . Lemma 1.5 (Fang [2]). Q = {Qx |x ∈ pt (I X )} defined in Definition 1.4 satisfies: (1) Qx (1X ) = 1 and Qx (0X ) = 0; (2) Qx (U ) > 0 ⇒ x qU ; (3) Qx (U ∧ V ) = Qx (U ) ∧ Qx (V );
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(4) Qx (U ) = y qV Qy (V ); x qV U (5) (U ) = x qU Qx (U ) for all U ∈ I X . Definition 1.6 (Peeters [10], Rodabaugh [14]). Let (X, ) be an I-fuzzy topological space and ∅ = Y ⊆ X. (Y, |Y ) is called the subspace of (X, ), where |Y : I Y → I is defined by (|Y )(U ) = {(V )|V ∈ I X , V |Y = U }. Theorem 1.7. Let f : (X, ) → (Y, ) be a continuous map and ∅ = Z ⊆ X. Then (f |Z) : (Z, |Z) → (Y, ) is continuous, where f |Z : Z → Y is defined by (f |Z)(x) = f (x) for x ∈ Z. Proof. Omitted. Definition 1.8. Let x ∈ pt (I X ) and S be a fuzzy net in I-fuzzy topological space (X, ) . The degrees to which S converges to x and S clusters to x are given as follows, respectively: Con(S, x ) =
{1 − Qx (U )| S is often ¬qU }
and Cl(S, x ) =
{1 − Qx (U )| S is finally ¬qU }.
Definition 1.9. Let (X, ) be an I-fuzzy topological space. If there exists a fuzzifying topology on X such that (A) = r∈I (r (A)) for all A ∈ I X , where r (A) = {x|A(x) > r}, then is called an induced I-fuzzy topology on X, denoted by (), i.e., = (). The pair (X, ) is called the induced I-fuzzy topological space of (X, ). Definition 1.10 (Shen [15]). Let (X, ) be a fuzzifying topological space. The degrees to which (X, ) is T0 , T1 and T2 defined as follows, respectively: ⎧⎛ ⎫ ⎞ ⎬ ⎨ ⎝ T0 (X, ) = Nx (A)⎠ ∨ Ny (B) x, y ∈ X, x = y ; ⎩ ⎭ y ∈A / x ∈B / ⎧⎛ ⎫ ⎞ ⎬ ⎨ ⎝ T1 (X, ) = Nx (A)⎠ ∧ Ny (B) x, y ∈ X, x = y ; ⎩ ⎭ y ∈A / x ∈B / T2 (X, ) = (Nx (A) ∧ Ny (B)) x, y ∈ X, x = y . A∩B=∅
2. T0 , T1 and T2 separation axioms The purpose of this section is to introduce q-T0 , s-T0 , T0 , T1 and T2 separation axioms on fuzzy points in I-fuzzy topological spaces. We will give their properties as well as the relations between them. Definition 2.1. Let (X, ) be an I-fuzzy topological space. (1) The degree to which two distinguished fuzzy points x and x with the same support point are quasi-T0 is defined as follows: ⎛ ⎞ ⎛ ⎞ q-T0 (x , x ) = ⎝ Qx (U )⎠ ∨ ⎝ Qx (V )⎠ . x ¬qU
x ¬qV
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The degree to which (X, ) is quasi-T0 is defined by q-T0 (X, ) = {q-T0 (x , x )|x ∈ X, = }. (2) The degree to which two distinguished fuzzy points x and y with x = y are T0 is defined as follows: ⎛ ⎞ ⎛ ⎞ T0 (x , y ) = ⎝ Qy (U )⎠ ∨ ⎝ Qx (V )⎠ . x ¬qU
y ¬qV
The degree to which (X, ) is T0 is defined by T0 (X, ) = {T0 (x , y )|x , y ∈ pt (I X ), x = y}. The degree to which (X, ) is sub-T0 is defined by ⎧ ⎫ ⎬ ⎨ s-T0 (X, ) = T0 (x , y )|x = y . ⎩ ⎭ ∈I0
(3) The degree to which two distinguished fuzzy points x and y with x = y are T1 is defined as follows: ⎛ ⎞ ⎛ ⎞ T1 (x , y ) = ⎝ Qy (U )⎠ ∧ ⎝ Qx (V )⎠ . x ¬qU
y ¬qV
The degree to which (X, ) is T1 is defined by T1 (X, ) = {T1 (x , y )|x , y ∈ pt (I X ), x = y}. (4) The degree to which two distinguished fuzzy points x and y with x = y are T2 is defined as follows: T2 (x , y ) = (Qy (V ) ∧ Qx (U )). U ∧V =0X
The degree to which (X, ) is T2 is defined by T2 (X, ) = {T2 (x , y )|x , y ∈ pt (I X ), x = y}. If is a Chang I-topology on X , then the concepts in Definition 2.1 coincide with the corresponding ones introduced in [9] if we emphasize the difference between the support points of fuzzy points. The following inequality holds: s-T0 (X, )T0 (X, ) T1 (X, ) T2 (X, ). Firstly, we see two examples. Example 2.2. Let X ⎧ ⎨ 1/4 (A) = 1/5 ⎩ 1
= {x, y} and : I X → I be defined as follows: A(x) > A(y), A(x) < A(y), A(x) = A(y).
Then is an I-fuzzy topology on X. From Definition 2.1 above, we have T0 (x1/2 , y1/2 ) = Furthermore, we can get T0 (x , y ) 41 for all , ∈ I0 . Then T0 (X, ) > T1 (X, ).
1 4
and T1 (x1/2 , y1/2 ) = 15 .
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Example 2.3. Let X be any infinite set and : I X → I be defined as follows: ⎧ A = 0X , 1X , ⎨1 (A) = 21 A satisfying {x|A(x) = 0} is a finite set, ⎩ 0 others. Then is an I-fuzzy topology on X. One can have that T1 (X, ) =
1 2
and T2 (X, ) = 0. Hence T1 (X, ) > T2 (X, ).
Theorem 2.4. Let (X, ) be an I-fuzzy topological space. Then (1) q-T0 (x , x ) = Qx ((x ) ) ( < ); (2) T0 (x , y ) = Qy ((x ) ) ∨ Qx ((y ) ); (3) T1 (x , y ) = Qy ((x ) ) ∧ Qx ((y ) ). Proof. From the definition of Qx , it is easy to prove it and omitted.
Theorem 2.5. Let (Y, |Y ) be the subspace of (X, ). (1) For any two distinguished fuzzy points x and x in (Y, |Y ), we have q-T0Y (x , x ) = q-T0X (x , x ). Hence q-T0 (X, )q-T0 (Y, |Y ). (2) For any two distinguished fuzzy points x and y with x = y in (Y, |Y ), we have T0Y (x , y ) = T0X (x , y ). Hence s-T0 (X, )s-T0 (Y, |Y ) and T0 (X, ) T0 (Y, |Y ). (3) For any two distinguished fuzzy points x and y with x = y in (Y, |Y ), we have T1Y (x , y ) = T1X (x , y ). Hence T1 (X, )T1 (Y, |Y ). (4) T2 (X, )T2 (Y, |Y ). Proof. We only prove (2) and the others can be obtained in a similar way and omitted. For any two distinguished fuzzy points x and y with x = y in I Y , from Theorem 2.4, we have T0Y (x , y ) = QYy ((x ) ) ∨ QYx ((y ) ), and X T0X (x , y ) = QX y ((x ) ) ∨ Qx ((y ) ). Y In order to prove that T0Y (x , y ) = T0X (x , y ), it suffices to show that QYx ((y ) ) = QX x ((y ) ) and Qy ((x ) ) = X Qy ((x ) ). In fact,
QYx ((y ) ) =
x qW (y )
|Y (W ) =
x qW (y ) V |Y =W
(V ) =
x qU (y )
(U ) = QX x ((y ) ).
Similarly, we can get QYy ((x ) ) = QX y ((x ) ), as desired. Naturally, we have s-T0 (X, ) s − T0 (Y, |Y ) and T0 (X, ) T0 (Y, |Y ).
Example 2.6. Let X = {x, y, z} and : I X → I be defined as follows: ⎧ 9/10 A(x) = A(y) > A(z), ⎪ ⎪ ⎪ ⎪ 8/10 A(x) = A(z) > A(y), ⎪ ⎪ ⎪ ⎪ ⎨ 7/10 A(y) = A(z) > A(x), (A) = 8/10 A(x) > A(y) and A(x) > A(z), ⎪ ⎪ 7/10 A(z) > A(x) and A(z) > A(y), ⎪ ⎪ ⎪ ⎪ 7/10 A(y) > A(x) and A(y) > A(z), ⎪ ⎪ ⎩ 1 others.
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Then is an I-fuzzy topology on X. Let Y = {y, z} ⊆ X. Then we can obtain that ⎧ ⎨ 9/10 B(y) > B(z), |Y (B) = 8/10 B(y) < B(z), ⎩ 1 B(y) = B(z). It is easy to verify that T1 (X, ) = 7/10 < 8/10 = T1 (Y, |Y ). Example 2.7. In Example 2.3, let Y = {x, y} ⊆ X. We can get |Y (B) = 1 for all B ∈ I Y . Hence T2 (Y, |Y ) = 1 > 0 = T2 (X, ). Theorem 2.8. Let (X, ) be an I-fuzzy topological space. Then T1 (X, ) = {((x ) )|x ∈ X, ∈ I0 }. Proof. For any two fuzzy points x , y with x = y, from (5) of Lemma 1.5, we have ⎫ ⎧ ⎬ ⎨ {((z ) )|z ∈ X, ∈ I0 } = Qw ((z ) )|z ∈ X, ∈ I0 ⎭ ⎩ w q(z ) Qw ((x ) ) w q(x )
Qy ((x ) ).
Similarly, we have {((z ) )|z ∈ X, ∈ I0 } Qx ((y ) ). Then {((z ) )|z ∈ X, ∈ I0 } T1 (x , y ). Hence
{((z ) )|z ∈ X, ∈ I0 } T1 (X, ).
On the other hand, we have T1 (X, ) = {Qy ((x ) ) ∧ Qx ((y V ) )|, ∈ I0 , x = y} {Qy ((x ) )|, ∈ I0 , x = y} ⎧ ⎫ ⎬ ⎨ = Qy ((x ) )| ∈ I0 , x ∈ X ⎩ ⎭ y ∈pt (I X ),y=x ⎧ ⎫ ⎬ ⎨ Qy ((x ) )| ∈ I0 , x ∈ X ⎩ ⎭ y q(x ) = {((x ) )| ∈ I0 , x ∈ X}. So T1 (X, ) = {((x ) )|x ∈ X, ∈ I0 }. Remark 2.9. In [6], Kotzé introduced T1 in (L, M)-topological spaces. An (L, M)-topological spaces (X, L, M, ) is T1 if and only if the map : LX → M assigns to the complement of a fuzzy point on X the value 1 in M. The following example will show that T1 defined in this paper is different from Kotzé’s. Example 2.10. Let X ⎧ ⎨ 9/10 (A) = 8/10 ⎩ 1
= {a, b}, L = M = I and be defined by A(a) > A(b), A(a) < A(b), A(a) = A(b).
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Then it is easy to verify that is an I-fuzzy topology on X. We can get T1 (X, ) = 8/10 according to our definition of T1 . Since ((a1/2 ) ) = 8/10 = 1, we know that (X, I, I, ) is not T1 in W. Kotzé’s sense. From this example, we know that what we give in this paper is the degree to which one I-fuzzy topological space is T1 and the value of the degree is in [0,1], which is very different from Kotzé’s. Question 2.11. In paper [6], Kotzé also introduced sober and semi-sober in (L, M)-topological spaces. How to introduce the corresponding sober and semi-sober in I-fuzzy topological spaces and discuss the relations with our T0 , T1 and T2 ? In [12,13], Rodabaugh defined a standard T0 axiom (denoted by RT0 in this paper) in L-topological spaces as follows: RT0 axiom: ∀x, y ∈ X such that x = y, there exists an open set u such that u(x) = u(y). In [8], Kubiak defined T1 and T2 axioms (denoted by KT1 and KT2 , respectively, in this paper) in L-topological spaces as follows: (1) KT1 axiom: If x = y, there exist two open set u, v such that u(x)u(y) and v(y)v(x). (2) KT2 axiom: If x = y, there exist two open set u, v such that u(x)u(y), v(y)v(x) and uv . If we replace the value lattice I = [0, 1] with {0, 1} in Definition 1.3, i.e., : I X → {0, 1} (or is a Chang I-topology), the readers can easily verify that the definitions of T0 , T1 and T2 defined in this paper imply RT0 , KT1 and KT2 , respectively. Besides this, we have more generalized result (See Theorem 2.12). Let (X, ) be an I-fuzzy topological space. We can rewritten Rodabaugh’s and Kubiak’s separation axioms for the I-Ftop setting as follows: (1) The degree to which two distinguished crisp points x, y ∈ X with x = y are RT0 is defined as follows: RT0 (x, y) = (u). u(x)=u(y)
The degree to which (X, ) is RT0 is defined by RT0 (X, ) = {RT0 (x, y)|x = y}. (2) The degree to which two distinguished crisp points x, y ∈ X with x = y are KT1 is defined as follows: KT1 (x, y) = (u) ∧ (v) = (u) ∧ (v). u(x)u(y)
v(y)u(x)
u(x)>u(y)
v(y)>v(x)
The degree to which (X, ) is KT1 is defined by KT1 (X, ) = {KT1 (x, y)|x = y}. (3) The degree to which two distinguished crisp points x, y ∈ X with x = y are KT2 is defined as follows: KT2 (x, y) = (u) ∧ (v) = (u) ∧ (v), u(x)u(y), v(y)v(x), u v
(u,v)∈D (x,y)
where D(x, y) = {(u, v)|u(x) > u(y), v(y) > v(x), uv }. The degree to which (X, ) is KT2 is defined by KT2 (X, ) = {KT2 (x, y)|x = y}. About the relation between above generalizations and ours, we have the following theorem. Theorem 2.12. Let (X, ) be an I-fuzzy topological spaces. The following formulas are valid: (1) s-T0 (X, )RT0 (X, ), (2) T1 (X, ) KT1 (X, ), (3) T2 (X, ) KT2 (X, ).
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Proof. We only prove (2) and (3). (2) Let t < T1 (X, ). On account of t < T1 (X, )T1 (x1 , y1 ) = Qy1 ((x1 ) ) ∧ Qx1 ((y1 ) ) =
y1 qB (x1 )
(B) ∧
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(A)
x1 qA (y1 )
for all x = y, there exists A ∈ I X such that x1 qA (y1 ) and t (A), and there exists B ∈ I X such that y1 qB (x1 ) and t (B), respectively. From x1 qA (y1 ) and y1 qB (x1 ) , we have A(x) > 0 = A(y) and B(y) > 0 = B(x). Hence t u(x)>u(y) (u) ∧ v(y)>v(x) (v). Therefore, t KT1 (X, ). From the arbitrariness of t, we have T1 (X, )KT1 (X, ), as desired. (3) Let t < T2 (X, ). Then t < A∧B=0X Qy1 (B) ∧ Qx1 (A) for all x = y . Hence there exist A, B ∈ I X such that A ∧ B = 0X and t < Qy1 (B) ∧ Qx1 (A). Furthermore, there exists C ∈ I X such that x1 qC A and t (C). Similarly, there exists D ∈ I X such that y1 qD B and t (D). By x1 qC A, y1 qD B and A ∧ B = 0X , we have C(x) > C(y), D(y) > D(x) and C D . Hence t (C)∧(D) (u,v)∈D(x,y) (u)∧(v). Therefore, t KT2 (X, ). Thus the conclusion. Remark 2.13. One thing we want to point out that the separation axioms of Rodabaugh’s and Kubiak’s are defined on crisp points and what we do in this paper is on fuzzy points. Besides the “separation degree’’ of points in background, we must also consider the same problem for points in different layers at the same time. This can be also seen from the proof of Theorem 2.12. When we prove Theorem 2.12, we obtain the results only using the fuzzy points with height 1. Now we see another example. Example 2.14. We consider the I-fuzzy unit interval I (I ) in I-topological spaces. For details about I (I ), please refer to [8,9]. It can be also regarded as I-fuzzy topology according to the characteristic function, i.e., 1 A ∈ I (I ), I (I )(A) = 0 A∈ / I (I ). We will show that q-T0 (X, I (I )) = T0 (X, I (I )) = 0. Hence T1 (X, I (I )) = T1 (X, I (I )) = 0. Firstly, we have q-T0 (X, I (I )) = 0. Let x ∈ X be defined as follows: ⎧ t < 0, ⎨ 1 x(t) = 1/2 0 t 1, ⎩ 0 t > 1. Then q-T0 (X, I (I ))q-T0 (x2/3 , x3/4 ) = Qx3/4 ((x2/3 ) ) = x3/4 qA (x2/3 ) I (I )(A) = 0. Secondly, we have T0 (X, I (I )) = 0. Let x, y ∈ X be defined as follows: ⎧ t < 0, ⎨ 1 x(t) = 1/3 0 t 1, ⎩ 0 t > 1. ⎧ t < 0, ⎨ 1 y(t) = 1/2 0 t 1, ⎩ 0 t > 1. Then T0 (X, I (I )) T0 (x1/3 , y1/3 ) = Qy1/3 ((x1/3 ) ) ∨ Qx1/3 ((y1/3 ) ) = I (I )(U ) ∨ y1/3 qU (x1/3 )
= 0.
x1/3 qV (y1/3 )
I (I )(V )
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Finally, we have T1 (X, I (I )) = T2 (X, I (I )) = 0. It is easy to verify that s-T0 (X, I (I )) = RT0 (X, I (I )) = KT1 (X, I (I )) = KT2 (X, I (I )) = 1. This is to say the degree to which I (I ) is KT2 (resp. KT1 , RT0 and s-T0 ) is 1 or I (I ) is KT2 (resp. KT1 , RT0 and s-T0 ). Remark 2.15. From the above example, we know that the degrees to which I (I ) is q-T0 , T0 , T1 and T2 are 0. So the I-fuzzy unit interval is not compatible with the T0 , T1 and T2 separation axioms proposed in this paper. It is an open to construct a model of R for I-Ftop and construct over separation axioms for I-Ftop so that this model is compatible (at least to some degree) with these axioms. Theorem 2.16. If (X, ) is an induced I-fuzzy topological space, i.e., there exists one fuzzifying topology on X such that = (), then (1) s-T0 (X, ) = T0 (X, ) = T0 (X, ), (2) T1 (X, ) = T1 (X, ), (3) T2 (X, ) = T2 (X, ). Proof. We prove (1) and (3), the proof of (2) is similar to that of (1). (1) At first, we want to prove T0 (X, ) T0 (X, ), this is to say T0 (X, ) Qy ((x ) ) ∨ Qx ((y ) ) for any two fuzzy points x , y with x = y. Let x , y ∈ pt (I X ) such that x = y and < T0 (X, ). On account of x = y, we have < Nx (X − {y}) or < Ny (X − {x}). No loss of generality, suppose < Nx (X − {y}), then there exists C ⊆ X such that x ∈ C ⊆ X − {y} and (C). Since (X, ) is induced I-fuzzy topological space, we have (C) = ( C ) Qx ( C )Qx ((y ) ), where C is the characteristic function of C. Hence Qy ((x ) ) ∨ Qx ((y ) ). Thus T0 (X, ). Therefore, T0 (X, ) T0 (X, ). On the other hand, we need to prove that s-T0 (X, ) Nx (X − {y}) ∨ Ny (X − {x}) for any x, y ∈ X with x = y. Let < s-T0 (X, ). Then for any x, y ∈ X with x = y, there exists ∈ I0 such that < T0 (x , y ). Hence < Qx ((y ) ) X or < Qy ((x ) ). For convenience, we assume that < Qx ((y ) ). Then there exists H ∈ I such that x qH (y ) and < (H ) = r∈I (r (H )). Let A = 1− (H ). Then we have x ∈ A, y ∈ / A and < (A) Nx (X − {y}). Thus, T0 (X, ). So s-T0 (X, ) T0 (X, ). (3) Let x, y be any two points in X such that x = y and let < T2 (X, ). Then there exist U, V ∈ I X with X U ∧ V = 0X such that < Qx1 (U ) and < Qy1 (V ). Then there exist W, H ∈ I with x1 qW U and y1 qH V such that < (W ) = r∈I (r (W )) and < (H ) = r∈I (r (H )). Now let A = 0 (W ) and B = 0 (H ). Then x ∈ A, y ∈ B, A∩B = ∅, < (A) qNx (A) and < (B) Ny (B). Thus T2 (X, ). Hence T2 (X, ) T2 (X, ). Conversely, let x , y be any two fuzzy points with x = y and let < T2 (X, ). Since x = y, there exist A, B ⊆ X with A ∩ B = ∅ such that < Nx (A) and < Ny (B). By < Nx (A), there exists C ⊆ X such that x ∈ C ⊆ A and < (C). By < Ny (B), there exists D ⊆ X such that y ∈ D ⊆ B and < (D). It is easy to check that x q C , y q D and C ∧ D = 0X . Since (X, ) is induced I-fuzzy topological space, we have (C) = ( C ) Qx ( C ) and (D) = ( D )Qy ( D ). Hence Qx ( C ) ∧ Qy ( D ). Therefore, T2 (X, ), as desired. Theorem 2.17. If f : (X, ) → (Y, ) is a homeomorphism. Then (1) q-T0 (x , x ) = q-T0 (f → (x ), f → (x )); (2) T0 (x , y ) = T0 (f → (x ), f → (y )); (3) T1 (x , y ) = T1 (f → (x ), f → (y )); (4) T2 (x , y ) = T2 (f → (x ), f → (y )). Proof. Omitted. Theorem 2.18. Let(X, ) be an I-fuzzy topological space. Then T2 (X, ) = {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in I X }. Proof. Let T2 (X, ) < . Then there exist two fuzzy points x , y with x = y such that Qx (V ) < or Qy (U ) < when U ∧ V = 0X . Let (Qx ) = {A|Qx (A) } and (Qy ) = {B|Qy (B) }. It is easy to prove that (Qx ) ×
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(Qy ) is a co-directed set. Furthermore, if A ∈ (Qx ) and B ∈ (Qy ) , then A ∧ B = 0X . Hence there exists x(A,B) ∈ X such that (A ∧ B)(x(A,B) ) > 0. Define S ∗ : (Qx ) × (Qy ) → pt (I X ) by S ∗ (A, B) = (x(A,B) )1−1/2[(A∧B)(x(A,B) )] for (A, B) ∈ (Qx ) × (Qy ) , then S ∗ is a fuzzy net. Now we want to show 1 − Con(S ∗ , x ) ∧ Con(S ∗ , y ) , i.e., {Qx (A)|S ∗ is often¬qA} ∨ {Qy (B)|S ∗ is often¬qB} . Suppose A ∈ I X such that S ∗ is often ¬qA. If Qx (A) , then A ∈ (Qx ) . Take B ∈ (Qy ) , then when (E, F ) ∈ (Qx ) × (Qy ) and (E, F ) ⊆ (A, B), we have, A(x(E,F ) ) + S ∗ ((E, F ))(x(E,F ) ) = A(x(E,F ) ) + 1 − 1/2(E ∧ F )(x(E,F ) ) > A(x(E,F ) ) + 1 − (E ∧ F )(x(E,F ) ) A(x(E,F ) ) + 1 − (A ∧ B)(x(E,F ) ) 1. Hence S(E, F )qA, i.e., S ∗ is eventually qA. This is contradict to the supposition. Therefore, Qx (A) < . Hence {Qx (A)| S is often ¬qA} . Similarly, {Qy (B)|Sis often¬qB} , as desired. Then {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . From the arbitrariness of , we have T2 (X, ) {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X}. Conversely, if T2 (X, ) > , then there exists U, V ∈ I X with U ∧ V = 0X such that Qx (U ) > and Qy (V ) > for any two fuzzy points x , y with x = y. For any fuzzy net S, if Con(S, x ) 1 − , then Qx (U ) when S is often ¬qU . Now we assert that {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . If not, then there exist a fuzzy net S and x , y with x = y such that 1 − Con(S, x ) ∧ Con(S, y ) , i.e., Con(S, x )1− and Con(S, y ) 1− . From Con(S, x ) 1− and Qx (U ) > , we know that S is eventually qU . Similarly, S is eventually qV . Hence S is eventually q(U ∧ V ). So U ∧ V = 0X , this is a contradiction. Therefore, {1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X} . Hence T2 (X, )
{1 − Con(S, x ) ∧ Con(S, y )| x , y ∈ pt (I X ), x = y, S is a fuzzy net in X}.
Thus the conclusion.
3. Separation axioms in the product space In this section,we study separation axioms in the product space. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces and P : j ∈J Xj → X be the projection. Then the product I-fuzzy topology whose subbase is defined by
∀W ∈ I j ∈J
Xj
, (W ) =
j ∈J
Pj← (U )=W
j (U ),
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and this product topology is denoted by j ∈J j . We know that if X (y) is a stratified factor space of ( ∈J X , ∈J ) parallel to X through y = (y ) ∈J , then (P |X (y))→ : (X (y), ∈J |X (y)) → (X , ) is a homeomorphism. Theorem 3.1. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces. Then (1) j ∈J q-T0 (Xj , j ) q-T0 ( j ∈J Xj , j ∈J j ); (2) j ∈J T0 (Xj , j )T0 ( j ∈J Xj , j ∈J j ); (3) j ∈J T1 (Xj , j )T1 ( j ∈J Xj , j ∈J j ); (4) j ∈J T2 (Xj , j )T2 ( j ∈J Xj , j ∈J j ). In addition, if (Xj , j ) is stratified for all j ∈ J , then the following equalities are true: (5) j ∈J q-T0 (Xj , j ) = q-T0 ( j ∈J Xj , j ∈J j ); (6) j ∈J T0 (Xj , j ) = T0 ( j ∈J Xj , j ∈J j ); (7) j ∈J T1 (Xj , j ) = T1 ( j ∈J Xj , j ∈J j ); (8) j ∈J T2 (Xj , j ) = T2 ( j ∈J Xj , j ∈J j ). Proof. We prove (3) and(4). (3) Let x , y ∈ pt (I j ∈J Xj ) be any two fuzzy points with x = y and let r ∈ I0 be any number with the property r< T1 (Xj , j ). Clearly, r < T1 (Xj , j ) for each j ∈ J . Due to x = y, there exists k ∈ J such that Pk (x) = Pk (y). j ∈J
We denote Pk (x), Pk (y) by a and b, respectively. Hence a , b ∈ pt (I Xk ) with a = b. Therefore, r < Qb ((a ) ) ∧ Qa ((b ) ). Then there exists C, D ∈ I Xk with b qC (a ) and a qD (b ) such that r < k (C) and r < k (D). Thus x qPk← (D)(y ) and y qPk← (C) (x ) . So j (Pk← (C))Qy (Pk← (C))Qy ((x ) ), r < k (C) j ∈J
and r < k (D)
j (Pk← (D))Qy (Pk← (D))Qx ((y ) ).
j ∈J
This means that
⎛ ⎞ Xj , Xj ⎠ . r {T1 (x , y )|x , y ∈ pt (I j ∈J Xj ), x = y} = T1 ⎝ j ∈J
j ∈J
From the arbitrariness of r, we get that ⎛ ⎞ T1 (Xj , j )T1 ⎝ Xj , j ⎠ . j ∈J
(4) Let hs , gt ∈ pt (I
j ∈J
j ∈J
Xj
) be any two fuzzy points with h = g and let r < ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ∀j ∈ J, r < Qxs (Aj ) ∧ Qyt (Bj )|s, t ∈ I0 , x = y . ⎪ ⎪ ⎩Aj ∧Bj =0X ⎭ j ∈J
j
j ∈J
T2 (Xj , j ). Then we have,
Y. Yue, J. Fang / Fuzzy Sets and Systems 157 (2006) 780 – 793
791
Since h = g, there exists j ∈ J such that h(j ) = g(j ). And then there exist G, H ∈ I Xj with G ∧ H = 0Xj such that h(j )s qG, g(j )t qH , j (G) > r and j (H ) > r. On account of Pj← (G) ∧ Pj← (H ) = 0 Xj , we can get ⎛
Qhs (Pj← (G)) ⎝
j ∈J
⎞ j ⎠ (Pj← (G))j (G) > r.
j ∈J
Similarly, Qgt (Pj← (G)) > r. Hence (Qhs (A) ∧ Qgt (B))Qhs (Pj← (G)) ∧ Qgt (Pj← (G)) > r. A∧B=0j ∈J
Xj
Therefore,
⎛
T2 (Xj , j )T2 ⎝
j ∈J
Xj ,
j ∈J
⎞ j ⎠ .
j ∈J
In addition, assume that (X j , j ) isstratified for j ∈ J , then (Xj , j ) is homeomorphic to some subspace (Xj (y), j ∈J j |Xj (y)) of ( j ∈J Xj , j ∈J j ). From Theorems 2.5 and 2.17, we have ⎛ ⎞ Xj , j ⎠ T2 (Xj (y), j |Xj (y)) = T2 (Xj , j ). T2 ⎝ j ∈J
j ∈J
j ∈J
j ∈J
j ∈J
Similarly, we have ⎛ ⎞ Xj , j ⎠ T1 (Xj (y), j |Xj (y)) = T1 (Xj , j ), T1 ⎝ j ∈J
⎛ T0 ⎝
j ∈J
Xj ,
j ∈J
and
⎛ q-T0 ⎝
j ∈J
⎞ j ⎠
j ∈J
Xj ,
j ∈J
Thus the conclusion.
j ∈J
T0 (Xj (y),
j ∈J
j ∈J
⎞ j ⎠
j ∈J
j |Xj (y)) =
j ∈J
q-T0 (Xj (y),
j ∈J
T0 (Xj , j )
j ∈J
j |Xj (y)) =
j ∈J
q-T0 (Xj , j ).
j ∈J
Example 3.2. Let X = {x} and Y = {y}. Define 1 : I X → I by 1 A ∈ {0X , 1X }, 1 (A) = 0 others. and define 2 : I Y → I by 2 (A) = 1 for all A ∈ I Y , respectively. Then 1 is an I-fuzzy topology on X and 2 is an I-fuzzy topology on Y. We can also calculate that the product topology 1 × 2 of 1 and 2 satisfies 1 × 2 (A) = 1 for all A ∈ I X×Y . It is easy to verify that q-T0 (X, 1 ) = 0, q-T0 (Y, 2 ) = 1 and q-T0 (X × Y, 1 × 2 ) = 1. Hence q-T0 (X, 1 ) ∧ q-T0 (Y, 2 ) < q-T0 (X × Y, 1 × 2 ). Theorem 3.3. Let {(Xj , j )}j ∈J be a family of I-fuzzy topological spaces. Then ⎛ ⎞ s-T0 (Xj , j ) = s-T0 ⎝ Xj , j ⎠ . j ∈J
j ∈J
j ∈J
792
Y. Yue, J. Fang / Fuzzy Sets and Systems 157 (2006) 780 – 793
Proof. Let r < S-T0 ( j ∈J Xj , j ∈J j ) and j0 ∈ J . For any two distinguished points a, b ∈ Xj0 , take x = (xj )j ∈J ∈ j ∈J Xj and y = (yj )j ∈J ∈ j ∈J Xj such that xj0 = a, yj0 = b and xj = yj whenever j ∈ J − {j0 }. Then there exists ∈ I0 such that r < T0 (x , y ). Hence ( ) (Al ) r < Qy ((x ) ) = y qG (x )
or r < Qx ((y ) ) =
l∈L
Al =G l∈L
x qH (y )
m∈M
( ) (Bm ).
Am =H m∈M
For convenience, we assume that r < Qy ((x ) ). Then there exist Gx (x ) and a family {Axl }l∈L such that y qGx = ( ) x x Al and r < (Al ). Furthermore, there exists l ∈ L such that y qAxl and r < ( ) (Axl ). On account of
l∈L
l∈L
( ) (Axl ) =
j (S),
( )n∈Ll Cn =Axl n∈Ll j ∈J Pj← (S)=Cn
there exists a finite family of {Cn }n∈Ll such that ( )n∈Ll Cn = Axl , and ∀n ∈ Ll , there exist Sjn ∈ I Xj such that (Sjn ) P ← (Sjn ) = Cn and jn (Sjn ) > r; Considering x ¬qAxl and y qAxl , then there exists n ∈ Ll such that x ¬qPj← n ← (S )(x) = P ← (S )(y). By the definition of x and y, we can assert that j = j . and y qPj← (S ). Therefore, P j j j n 0 n n n jn jn n Hence r < jn (Sjn )Qb (Sjn ) Qb ((a ) ). Thus, s-T0 (Xj0 , j0 ) =
⎧ ⎨ ⎩
T0 (x , y )|x = y
∈I0
⎫ ⎬ ⎭
r.
Therefore,
⎛ s-T0 (Xj , j ) s-T0 ⎝
j ∈J
The proof of
j ∈J
j ∈J
Xj ,
⎞ j ⎠ .
j ∈J
s-T0 (Xj , j ) s-T0 (
j ∈J
Xj ,
j ∈J
j ) is similar to that of (3) in Theorem 3.1.
Acknowledgements The authors would like to thank Prof. S.E. Rodabaugh and the anonymous referees for their very helpful comments and valuable suggestions.
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