A TYCHONOFF THEOREM IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

The purpose of this paper is to prove a Tychonoﬀ theorem in the so-called “intuitionis-tic fuzzy topological spaces.” After giving the fundamental deﬁnitions, such as the deﬁni-tions of intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic fuzzy topological space, fuzzy continuity, fuzzy compactness


Introduction.
After the introduction of the concept of fuzzy sets by Zadeh [8] several researches were conducted on the generalizations of the notion of fuzzy set.The idea of intuitionistic fuzzy set (IFS) was first published by Atanassov [1] and many works by the same author and his colleagues appeared in the literature [1,3,4].

2.
Preliminaries.First we will present the fundamental definitions obtained by Atanassov.
Definition 2.1 [1].Let X be a nonempty fixed set.An IFS A is an object having the form where the functions µ A : X → I and γ A : X → I denote the degree of membership (namely, µ A (x)) and the degree of nonmembership (namely, γ A (x)) of each element x ∈ X to the set A, respectively, and 0 ≤ µ A (x) + γ A (x) ≤ 1 for each x ∈ X.
Remark 2.2.An IFS A = { x, µ A (x), γ A (x) : x ∈ X} in X can be identified to an ordered pair µ A ,γ A in I X ×I X or to an element in (I × I) X .Remark 2.3.For the sake of simplicity, we will use the symbol A = x, µ A ,γ A for the IFS A = { x, µ A (x), γ A (x) : x ∈ X}.
Example 2.4.Every fuzzy set A on a nonempty set X is obviously an IFS having the form A = { x, µ A (x), 1 − µ A (x) : x ∈ X} [1].
One can define several relations and operations between IFSs as follows.
Definition 2.5 [2].Let X be a nonempty set, and let the IFSs A and B be in the form We can easily generalize the operations of intersection and union in Definition 2.5 to arbitrary family of IFSs as follows.
Definition 2.6.Let {A i : i ∈ J} be an arbitrary family of IFSs in X.Then (a) Definition 2.7.The ordinary complement of A = x, µ A ,γ A is defined by Definition 2.8.0 ∼ = { x, 0, 1 : x ∈ X} and 1 ∼ = { x, 1, 0 : x ∈ X}.Now we will define the image and preimage of IFSs.Let X and Y be two nonempty sets and let f : Now we generalize the concept of fuzzy topological space, first initiated by Chang [6], to the case of IFSs.
This construction, in some sense, has a close resemblance to that of Brown [5], the so-called "fuzzy ditopological space."Definition 2.9 [7].An intuitionistic fuzzy topology (IFT) on a nonempty set X is a family τ of IFSs in X satisfying the following axioms: ) ∪G i ∈ τ, for any arbitrary family {G i : i ∈ τ} ⊆ τ.
In this case the pair (X, τ) is called an intuitionistic fuzzy topological space (IFTS) and any IFS in τ is known as an intuitionistic fuzzy open set (IFOS) in X.
Example 2.10.Any fuzzy topological space (X, τ 0 ) in the sense of Chang is obviously an IFTS whenever we identify a fuzzy set A in X whose membership function is µ A with its counterpart as in Example 2.4.
Notice that any fuzzy topological space (X, τ) in the sense of Lowen is obviously an IFTS in the sense of Lowen.
Definition 2.15.The complement A of an IFOS A in an IFTS (X, τ) is called an intuitionistic fuzzy closed set (IFCS) in X. Definition 2.16.Let (X, τ) be an IFTS on X.(a) A subfamily β ⊆ τ is called a base for (X, τ), if each member of τ can be written as a union of elements in β.
(b) A subfamily S ⊆ τ is called a subbase for (X, τ), if the family of all finite intersections of S forms a base for (X, τ).In this case it is said that (X, τ) is generated by S. Now we define fuzzy closure and interior operators in an IFTS.Definition 2.17.Let (X, τ) be an IFTS and let A = { x, µ A (x), γ A (x) : x ∈ X} be an IFS in X.Then the fuzzy interior and fuzzy closure of A is defined by (2.5) It can also be shown that cl(A) is an IFCS and int(A) is an IFOS in X, and A is an IFCS in X if and only if cl(A) = A, A is an IFOS in X if and only if int(A) = A. In addition, for any IFS A in (X, τ), For further properties of fuzzy interior and closure operators you may consult [7].
where τ 1 and τ 2 are the fuzzy topological spaces on X defined in Example 2.13.Now we present the basic definitions concerning fuzzy continuity.

Intuitionistic fuzzy compactness.
First we present the basic concepts.Definition 3.1 [7].Let (X, τ) be an IFTS.(a) If a family { x, µ Gi ,γ Gi : i ∈ J} of IFOSs in X satisfies the condition ∪{ x, µ Gi , γ Gi : i ∈ J} = 1 ∼ , then it is called a fuzzy open cover of X.A finite subfamily of a fuzzy open cover { x, µ Gi ,γ Gi : i ∈ J} of X, which is also a fuzzy open cover of X, is called a finite subcover of { x, µ Gi ,γ Gi : i ∈ J}.
(c) An IFTS (X, τ) is called fuzzy compact if and only if every fuzzy open cover of X has a finite subcover.Now we will give two characterizations of fuzzy compactness.

Proposition 3.2. An IFTS (X, τ) is fuzzy compact if and only if for each family δ = {G
The proof of Proposition 3.2 is obvious.

Proposition 3.3. An IFTS (X, τ) is fuzzy compact if and only if every family { x, µ Ki , γ Ki : i ∈ J} of IFCSs on X having the FIP has a nonempty intersection.
For the proof of Proposition 3.3 see [7].Here we state that fuzzy compactness is preserved under a fuzzy continuous bijection.

Proposition 3.4. Let (X, τ), (Y , φ) be IFTSs and f : X → Y a fuzzy continuous bijection. If (X, τ) is fuzzy compact, then so is (Y , φ).
For the proof of Proposition 3.4 see [7].Definition 3.5 [7].Let (X, τ) be an IFTS and A an IFS in X.(a) If a family { x, µ Gi ,γ Gi : i ∈ J} of IFOSs in X satisfies the condition A ⊆ ∪{ x, µ Gi ,γ Gi : i ∈ J}, then it is called a fuzzy open over of A. A finite subfamily of the fuzzy open cover { x, µ Gi ,γ Gi : i ∈ J} of A, which is also a fuzzy open cover of A, is called a finite subcover of { x, µ Gi ,γ Gi : i ∈ J}.
(b) An IFS A = x, µ A ,γ A in an IFTS (X, τ) is called fuzzy compact if and only if every fuzzy open cover of A has a finite subcover.Corollary 3.6 [7].An IFS A = x, µ A ,γ A in an IFTS (X, τ) is fuzzy compact if and only if for each family δ = {G i : i ∈ J}, where G i = x, µ Gi ,γ Gi (i ∈ J) of IFOSs in X with the properties µ A ⊆ ∨ i∈J µ Gi and 1 − γ A ⊆ ∨ i∈J (1 − γ Gi ), there exists a finite subfamily Example 3.7.Let (X, τ 0 ) be a fuzzy topological space in Chang's sense and µ A ∈ I x a fuzzy compact set in X.We can construct an IFTS τ on X as in [7,Example 5.9].Now the IFS A = x, µ A , 1 − µ A , also fuzzy compact in the IFTS (X, τ).Corollary 3.8.Let (X, τ), (Y , φ) be IFTSs and f : X → Y a fuzzy continuous bijection.If A is fuzzy compact in (X, τ), then so is f (A) in (Y , φ).
For the proof of Corollary 3.8 see [7].Here we present a version of the Alexander subbase lemma for IFTSs.Proposition 3.9 (the Alexander subbase lemma).Let δ be a subbase of an IFTS (X, τ).Then (X, τ) is fuzzy compact if and only if for each family of IFCSs chosen from δ c = {K : K ∈ δ} having the FIP there is a nonzero intersection.
"Only if" part.Suppose, on the contrary, that (X, τ) is not fuzzy compact.Then there exists a family K = { x, µ Ki ,γ Ki : i ∈ J} of IFCSs in X having the FIP such that ∩ i∈J K i = 0 ∼ .The collection of all such families K, ordered by classical inclusion (⊆), is inductive.Hence, by Zorn's lemma, we may find a maximal family K, say M = {M j : j ∈ K}, where M j = x, µ Mj ,γ Mj , j ∈ K.The family M satisfies the following properties: (1) M is a fuzzy filter consisting of IFCSs in (X, τ), (2) ∩(M ∩ δ c ) = 0. Now the second assertion gives an immediate contradiction to the hypothesis of the theorem.Hence, (X, τ) is fuzzy compact.

A Tychonoff theorem in IFTSs.
With a Tychonoff-like theorem in mind, we must first present the product of IFTSs.Let (X i ,τ i ) be an IFTS on X i for each i ∈ J, and let X = πX i .For each i ∈ J, we may construct the ith projection mapping as follows: Then we define Definition 4.1.The product set X equipped with the IFT generated on X by the family S is called the product of the IFTSs {(X i ,τ i ) : i ∈ J}.
For each i ∈ J and for each S i ∈ τ i we have π −1 i (S i ) ∈ τ; so π i is indeed a fuzzy continuous function from the product IFTS onto (X i ,τ i ) for all i ∈ J.The product IFT τ is, of course, the coarsest IFT on X having this property.Now we express Tychonoff theorem in two steps, one of which is valid in general, but its reverse is true for finitely many terms.Theorem 4.2.For each i ∈ J let (X i ,τ i ) be an IFTS on X i .Then if the product IFTS (X, τ) is fuzzy compact, then each (X i ,τ i ) is fuzzy compact.
Proof.Here we will make use of the Alexander subbase lemma.Suppose, on the contrary, that there exists a family consisting of some of the IFCSs obtained from the subbase of the product IFT on X such that P has the FIP and ∩P = 0. Now it can be shown easily that the families have the FIP, and since (X i ,τ i )'s (i = 1, 2) are fuzzy compact, we have ∩P 1 ≠ 0 and ∩P 2 ≠ 0 which mean that But from ∩P = 0 we obtain Here there exist four cases.Case 1.If ∧µ Pi1 ≠ 0 and ∧µ Pi2 ≠ 0, then there exist x 1 ∈ X 1 , x 2 ∈ X 2 such that ∧µ Pi1 (x 1 ) ≠ 0 and ∧µ Pi2 (x 2 ) ≠ 0 from which we obtain a contradiction to (4.7), if it is evaluated in (x 1 ,x 2 ).
Case 2. If ∨γ Pi1 ≠ 1 and ∨γ Pi2 ≠ 1, then we get a similar contradiction as in the first case.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Definition 2 .
19.Let (X, τ) and (Y , φ) be two IFTSs and let f : X → Y be a function.Then f is said to be fuzzy continuous if and only if the preimage of each IFOS in φ is an IFOS in τ.Definition 2.20.Let (X, τ) and (Y , φ) be two IFTSs and let f : X → Y be a function.Then f is said to be fuzzy open if and only if the image of each IFS in τ is an IFS in φ.Example 2.21.Let (X, τ 0 ), (X, φ 0 ) be two fuzzy topological spaces in the sense of Chang.(a) If f : X → Y is fuzzy continuous in the usual sense, then in this case, f is fuzzy continuous in the sense of Definition 2.19, too.(b) If f : X → Y is fuzzy open in the usual sense, then f need not be fuzzy open in the sense of Definition 2.20.(If, furthermore, f : X → Y were both one-to-one and onto, then f would also be fuzzy open in the sense of Definition 2.20.)Example 2.22.Let (X, τ) be an IFTS in the sense of Lowen, (Y , φ) an IFTS, and c 0 ∈ Y .Then the constant function c : X → Y , c(x) = c 0 is obviously fuzzy continuous.Example 2.23.Let (Y , φ) be an IFTS, X a nonempty set, and f : X → Y a function.In this case τ