COMPACTNESS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

We introduce fuzzy almost continuous mapping, fuzzy weakly continuous mapping, fuzzy compactness, fuzzy almost compactness


Introduction and preliminaries
The concept of a fuzzy set was introduced by Zadeh [13], and later Chang [3] defined fuzzy topological spaces.These spaces and their generalizations are later studied by several authors, one of which, developed by Šostak [11,12], used the idea of degree of openness.This type of generalization of fuzzy topological spaces was later rephrased by Chattopadhyay et al. [4], and by Ramadan [10].
In this paper, we introduce the follwing concepts: fuzzy almost continuous mapping, fuzzy weakly continuous mapping, fuzzy compactness, fuzzy almost compactness, and fuzzy near compactness in intuitionistic fuzzy topological spaces in view of the definition of Šostak.
Definition 1.1 [1].Let X be a nonempty fixed set and I the closed unit interval [0,1].An intuitionistic fuzzy set (IFS) A is an object having the form where the mappings µ A : X → I and ν A : X → I denote the degree of membership (namely, µ A (x)) and the degree of nonmembership (namely, ν A (x)) of each element x ∈ X to the set A, respectively, and 0 ≤ µ A (x) + ν A (x) ≤ 1 for each x ∈ X.The complement of the IFS A, is A = { x,ν A (x),µ A (x) : x ∈ X}.Obviously, every fuzzy set A on a nonempty set X is an IFS having the form For a given nonempty set X, denote the family of all IFSs in X by the symbol ζ X .
Definition 1.2 [6].Let X be a nonempty set and x ∈ X a fixed element in X.If r ∈ I 0 , s ∈ I 1 are fixed real numbers such that r + s ≤ 1, then the IFS x r,s = y,x r ,1 − x 1−s is called an intuitionistic fuzzy point (IFP) in X, where r denotes the degree of membership of x r,s , s the degree of nonmembership of x r,s , and x ∈ X the support of x r,s .The IFP x r,s is contained in the IFS A (x r,s ∈ A) if and only if r < µ A (x), s > γ A (x).
Definition 1.3 [6].(i) An IFP x r,s in X is said to be quasicoincident with the IFS A, denoted by x r,s qA, if and only if r > γ A (x) or s < µ A (x). x r,s qA if and only if x r,s ∈ A.
(ii) The IFSs A and B are said to be quasicoincident, denoted by AqB if and only if there exists an element x ∈ X such that µ A (x) > γ B (x) or γ A (x) < µ B (x).If A is not quasicoincident with A, denote A qB.A qB if and only if A ⊆ B. Definition 1.5 [5].An intuitionistic fuzzy topology (IFT) in Chang's sense on a nonempty set X is a family τ of IFSs in X satisfying the following axioms: In this case, the pair (X,τ) is called Chang intuitionistic fuzzy topological space and each IFS in τ is known as intuitionistic fuzzy open set in X. Definition 1.6 [8].An IFS ξ on the set ζ X is called an intuitionistic fuzzy family (IFF) on X.In symbols, denote such an IFF in form ξ = µ ξ ,ν ξ .

A. A. Ramadan et al. 21
Definition 1.7 [7].An IFT in Šostak's sense on a nonempty set X is an IFF τ on X satisfying the following axioms: In this case, the pair (X,τ) is called an intuitionistic fuzzy topological space in Šostak's sense (IFTS).For any A ∈ ζ X , the number µ τ (A) is called the openness degree of A, while ν τ (A) is called the nonopenness degree of A.
Then, τ is an IFT in the sense of Šostak and neither a Chang fuzzy topology nor a Chang IFT.
Definition 1.9 [7].Let (X,τ) be an IFTS on X.Then the IFF τ * is defined by τ Theorem 1.10 [7].The IFF τ * on X satisfies the following properties: Definition 1.11 [7].Let (X,τ) be an IFTS and A be an IFS in X.Then the fuzzy closure and fuzzy interior of A are defined by where Theorem 1.12 [7].The closure and interior operator satisfy the following properties: Definition 1.13 [7].Let (X,τ 1 ) and (Y ,τ 2 ) be two IFTSs and f : X → Y be a mapping.Then f is said to be (i) intuitionistic fuzzy continuous if and only if
Proof.(i) Let A, B be any two (α,β)-IFRC sets.By Theorem 2.2, we have τ (ii) It can be proved by the same manner.
Theorem 2.4.Let (X,τ) be an IFTS.Then, (ii) It can be proved by the same manner.
Remark 2.6.From the above definition, it is clear that the following implications are true for α ∈ I 0 , β ∈ I 1 with α + β ≤ 1: (α,β)-intuitionistic fuzzy almost continuous mapping intuitionistic fuzzy strong continuous intuitionistic fuzzy continuous mapping (α,β)-intuitionistic fuzzy weakly continuous mapping (2.5) But, the reciprocal implications are not true in general, as shown by the following examples.
Example 2.9.In the above example, if then f is intuitionistic fuzzy continuous, but not intuitionistic fuzzy strong continuous.
Theorem 2.12.Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect to the IFTs τ 1 and τ 2 respectively.Then for every IFS A in X, where Proof.Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect to τ 1 and τ 2 , and let A ∈ ζ X .Then, (2.16) Theorem 2.13.Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect to the IFTs τ 1 and τ 2 , respectively.Then, for every IFS A in Y , where,
Proof.Let an IFTS (X,τ) be (α,β)-intuitionistic fuzzy compact.Then, for every family For the second implication, suppose that the IFTS (X,τ) is (α,β)-intuitionistic fuzzy nearly compact, then for every family Hence, the IFTS (X,τ) is (α,β)intuitionistic fuzzy almost compact.Remark 3.5.In IFTS in Chang's sense, the converse of these two implications are not valid for compactness, nearly compactness, and almost compactness [9], which are special cases of compactness, nearly compactness and almost compactness, respectively in IFTS, in Šostak's sense.Thus, the converse implications in Theorem 3.4 are not true in general.

intuitionistic fuzzy compact, if and only if every family in
where, α ∈ I 0 , β ∈ I 1 with α + β ≤ 1 having the FIP, has a nonempty intersection.
Theorem 3.11.An IFTS (X,τ) is (α,β)-intuitionistic fuzzy almost compact if and only if every family which is a contradiction with the FIP of the family.
Proof.Let x r,s ∈ cl α,β V and let U be any IFS in X such that τ(U) ≥ α,β and x r,s qU.Suppose for a contradiction that V qU.Then, we have V ⊆ U. Since x r,s qU, then x r,s ∈ U ⊇ V , and since τ * (U) = τ(U) ≥ α,β , then x r,s ∈ cl α,β V , which is a contradiction, then VqU.Conversely, suppose that for any U ∈ ζ X with τ(U) ≥ α,β such that x r,s qU, we have UqV.Suppose, for a contradiction, that x r,s ∈ cl α,β V .Then, there exists B ∈ ζ X with τ * (B) ≥ α,β , B ⊇ V and x r,s ∈ B. Thus, τ(B) = τ * (B) ≥ α,β and x r,s qB.Then, from our hypotheses VqB, which implies that V ⊆ B; this is a contradiction.Hence, x r,s ∈ cl α,β V .Lemma 3.13.Let (X,τ) be an IFTS.For Let x t,s ∈ cl α,β A, where r ∈ I 0 , s ∈ I 1 with r + s ≤ 1.Then by Lemma 3.12, there exists U ∈ ζ X with τ(U) ≥ α,β such that x t,s qU and A qU. From A qU, it follows that A ⊆ U, so using the fact that τ * (U) = τ(U) ≥ α,β , we obtain that cl α,β A ⊆ U.

Proof. Let {G
Then from the almost compactness of (X,τ 1 ), there exists a subset J 0 of J such that But from Theorem 2.13, we have Since (X,τ 1 ) is (α,β)-intuitionistic fuzzy compact, there exists a finite subset J 0 of J such that

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 1 . 4
[8].Let a and b be two real numbers in [0,1] satisfying the inequality a + b ≤ 1.Then the pair a,b is called an intuitionistic fuzzy pair.Let a 1 ,b 1 , a 2 ,b 2 be two intuitionistic fuzzy pairs.Then define (i) a 1 ,b 1 ≤ a 2 ,b 2 if and only if a 1 ≤ a 2 and b 1 ≥ b 2 ; (ii) a 1 ,b 1 = a 2 ,b 2 if and only if a 1 = a 2 and b 1 = b 2 ; (iii) if { a i ,b i : i∈J} is a family of intuitionistic fuzzy pairs, then ∨ a i ,b i = ∨a i ,∧b i and ∧ a i ,b i = ∧a i ,∨b i ; (iv) the complement of an intuitionistic fuzzy pair a,b is the intuitionistic fuzzy pair defined by a,b = b,a ; (v) 1 ∼ = 1,0 and 0 ∼ = 0,1 .