Intuitionistic fuzzy-γ-retracts and interval-valued intuitionistic almost ( near ) compactness

The aim of this paper is to introduce the concepts of an intuitionistic fuzzy-γ-retract and an intuitionistic fuzzy-Rretract. Some characterizations of these new concepts are presented. Examples are given, and properties are established. Also, we study the concepts of interval-valued intuitionistic almost (near) compactness and define S1-regular spaces. We prove that if an intuitionistic fuzzy topological space is an S1-regular space and interval-valued intuitionistic almost (near) compact, then it is interval-valued intuitionistic compact.


INTRODUCTION AND PRELIMINARIES
The concept of fuzzy sets was first proposed by Zadeh in 1965 [8].This concept has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics, and many more.It gives us a tool to model the uncertainty present in a phenomenon that does not have sharp boundaries.Many papers on fuzzy sets have been published, showing their importance and applications to set theory, algebra, real analysis, measure theory, topology, etc.
Atanassov [1] extends the fuzzy set characterized by a membership function to the intuitionistic fuzzy set (IFS), which is characterized by a membership function, a non-membership function, and a hesitancy function.As a result, the IFS can describe the fuzzy characters of things in more detail and more comprehensively, which is found to be more effective in dealing with vagueness and uncertainty.Over the last few decades, the IFS theory has been receiving more and more attention from researchers and practitioners, and has been applied to various fields, including decision making, logic programming, medical diagnosis, pattern recognition, robotic systems, fuzzy topology, machine learning, and market prediction.
Intuitionistic fuzzy sets as a generalization of fuzzy sets can be useful in situations when the description of a problem by a (fuzzy) linguistic variable, given in terms of a membership function only, seems too rough.For example, in decision-making problems, particularly in the case of medical diagnosis, sales _________________________ analysis, new product marketing, etc., there is a fair chance of the existence of a non-null hesitation part at each moment of evaluation of an unknown object.The concept of an intuitionistic fuzzy set, originally proposed by Atanassov [1], is an important tool for dealing with imperfect and imprecise information.Compared with Zadeh's fuzzy sets, an intuitionistic fuzzy set gives the membership and non-membership degree to which an element belongs to a set.Hence, coping with imperfect and imprecise information is more flexible and effective for intuitionistic fuzzy sets.In recent years, intuitionistic fuzzy set theory has been successfully applied in many practical fields, such as decision analysis and pattern recognition.Combining intuitionistic fuzzy set theory and rough set theory may be a promising topic that deserves further investigation.Some research has already been carried out on this topic.
In 1965, Zadeh presented the idea of a fuzzy sеt [8] as a means to represent uncertainty.This notion was originally introduced as a method to consider imprecision and ambiguity occurring in human discourse and thought.Many works by the same author and his colleagues appeared in the literature [3,4].Later, topological structures in fuzzy topological spaces [5] were generalized to intuitionistic fuzzy topological spaces by Coker in [4], who then introduced the concept of an intuitionistic set [4].This concept is the discrete form of an intuitionistic fuzzy set, and it is one of several ways of introducing vagueness in mathematical objects.On the other hand, the concept of a fuzzy retract was introduced by Rodabaugh [7].
The purpose of this paper is to construct the idea of intuitionistic fuzzy retracts, called IF-R-retracts, which use the gеnеralizatiοn of intuitionistic fuzzy continuity.After giving the fundamental examples, we introduce the concepts of interval-valued intuitionistic almost (near) compactness and S 1 -regular spaces and prove that if an intuitionistic fuzzy topological space ( , ) X  is an S 1 -regular space and interval-valued intuitionistic almost (near) compact, then it is an interval-valued intuitionistic compact.
Throughout this paper, X denotes a non-empty set.A fuzzy set in X is a function with domain X and values in I.The words intuitionistic fuzzy set and intuitionistic fuzzy topological space will be abbreviated as IF-set and IF-ts, respectively.Also, by I(ν), C(ν), and ν′ we will denote respectively the interior, closure, and the complement of an IF-set ν.A mapping r : (X , δ ) → ( , ) X  be an IF-ts and .

A X
 Then a maximal subspace ( A , δ A ) of ( , ) X  is an IF-ts and is defined by { : }. [1].Let X be a nonempty set.An IF-set A is an object of the form { , ( ), ( ): },   where the functions µ A : X → [0,1] and ν A : X → [0,1] denote, respectively, the degree of membership function (namely µ A (x)) and the degree of non-membership function (namely [2].Atanassova and Doukovska introduced the following interesting geometrical interpretations to express an IF-set (see Fig. 1).Definition 1.2 [1].Let X be a nonempty fixed set, and let I be the closed unit interval [0,1].Consider two IF-sets { , ( ), ( ): [6].Let A be an IF-set of an IF-ts ( , ). (ix) an IF-semi-precontinuous mapping if for each    we have f ← (ν) is an IF-semi-preopen set of X. Definition 1.5 [6].Let ( , ) X  be an IF-ts and let .

A X  Then the IF-subspace ( , )
A A  is called an IF- retract (IFR, for short) of ( , ) X  if there exists an IF-continuous mapping r : ( , ) X  → ( , ) A A  such that r(a)=a for all a A  .In this case r is called an IF-retraction.

X  be an IF-ts. Then ( , )
A A  is said to be an IF-neighbourhood semi-retract Definition 1.9 [6].Let f : ( , ) X  be an .
of interval-valued intuitionistic fuzzy sets ( , of s I  of X satisfies the finite intersection property (FIP, for short) iff every finite subfamily {λ j 1 ,λ j 2 ,λ j 3 ,…,λ j n } of M satisfies the condition X  is called interval-valued intuitionistic fuzzy compact ( , c I  for short) iff every o I  has a finite subcover.

INTUITIONISTIC FUZZY-γ-RETRACTS
In this section the basic concept of an intuitionistic fuzzy-γ-retract is introduced, and some characterizations are presented.Examples and properties are established.Also, the relations between these new concepts are explained.
X  be an IF-ts and A X  .Then a maximal subspace ( , ) From the above definitions one may notice that is an IF-strongly semiretract of ( , ) but not an IF-strongly semi-retract.
X  be an IF-ts and .5 0.5 0.4 0.5 0.5 0.6 : ) ) A A  is an IF-preretract of ( , ) X  but not an IF-semi-retract.

IF-R-CONTINUITY AND IF-R-RETRACTS
In this section the basic concepts of intuitionistic fuzzy perfectly retracts, intuitionistic fuzzy R retracts, and intuitionistic fuzzy completely retracts and some characterizations are discussed.Many examples are given, and some properties are established.Also, we define the relations between these new concepts.

and ( , )
Y  be two IF-ts's where 0.7,0.2 , X  be an IF-ts, A X  and let r : ( , ) be a mapping such that r(a)  a for all a A  .If the graph g : ( , ) retraction, where θ is the product topology generated by δ and δ A .
Proof.It follows directly from Theorem 3.3.
X  be an IF-ts.Then ( , ) A A  is said to be an IF-neighbourhood-perfectly retract (resp., IF-neighbourhood completely retract) (resp., IF-nbd PR, IF-nbd CR, for short) of ( , ) (i) Every IFPR is an IF-nbd PR, but the converse is not true.
(ii) Every IFCR is an IF-nbd CR, but the converse is not true.

Then ( , )
A A  is an IF-nbd CR of ( , ) X  but not an IFCR of ( , ), X  and it is an IF-nbd PR of ( , ) X  but not an IFPR of ( , ).X 

INTERVAL-VALUED INTUITIONISTIC COMPACTNESS
In this section we introduce the concepts of interval-valued intuitionistic almost (near) compactness and define S 1 -regular spaces.We prove that if ( , ) X  is an S 1 -regular space and interval-valued intuitionistic almost (near) compact, then it is interval-valued intuitionistic compact.
are IF-open sets of X and 1,  implies that λ is a finite subcover of X.Then the intuitionistic fuzzy topological space ( , ) X  is an intuitionistic fuzzy compact space.
X  be an c I  space, and let { , For the second implication, assuming ( , ) .
Y y is also .
Y y be a surjection and interval-valued intuitionistic fuzzy-weakly continuous.If ( , ) Since f is an IVIF-weakly continuous mapping, then we have ( ) ( ( )).
Y  be a surjection and interval-valued intuitionistic fuzzy-strongly continuous.If ( , ) Since f is IVIF-strongly continuous and hence a continuous mapping, then we have { ( ), } there exists a finite subfamily X  be an interval-valued intuitionistic fuzzy topological space.Then the following conditions are equivalent: it follows that there exists a finite subfamily

Example 2 . 4 .
Let λ be an IF-set on X = {a, b, c } defined by ,

Example 2 . 5 .
Let λ be an IF-set on X = {a, b, c } defined by ,

Example 2 . 6 .
Let λ be an IF-set on X = {a, b, c } defined by ,

Example 2 .
10. Let λ be an IF-set on X={a,b,c} defined by ,

Definition 3 . 1 .Remark 3 . 1 .
Let f : ( , ) X  → ( , ) Y  be a mapping from an IF-ts ( , ) X  to another IF-ts ( , ).Y  Then f is called (i) an IF-perfectly continuous (IFPC, for short) mapping if for each    we have f ← (ν) is both an IFopen and an IF-closed set of X, (ii) an IF-completely continuous (IFCC, for short) mapping if for each    we have f ← (ν) is an IFregular open set of X, (iii) an IF-R-continuous (IFRC, for short) mapping if for each IF-regular open    we have f ← (ν) is an IF-regular open set of X.The implications between these different concepts are given by the following diagram: IFPC  IFCC  IFRC.The converses of the above implications need not be true in general, as shown by the following examples.

Definition 3 . 2 .Lemma 3 . 1 .□Theorem 3 . 1 .□Theorem 3 . 2 .□Theorem 3 . 3 .
An IF-ts ( , ) X  is called an IF-extremely disconnected space (IFED-space, for short) if the closure of every IF-open set of X is an IF-open set.Let ( , ) X  be an IFED-space.Then, if λ is an IF-regular open set of X, it is both an IF-open set and an IF-closed set.Proof.Let λ be an IF-regular open set of X, then λ  I(C(λ)) since every IF-regular open set is IF-open.Then λ is an IF-open set of X and because ( , ) X  is an IFED-space, C(λ)  λ.Then λ is an IF-closed set.Let ( , ) X  be an IFED-space, and let f : ( , ) X  → ( , ) Y  be a mapping.Then the following are equivalent: (i) f is IFPC, (ii) f is IFCC.Proof.It follows from Lemma 3.1.Let f : ( , ) X  → ( , ) Y  be a mapping.Then f is IFPC (resp., IFCC) iff the inverse image of every IF-closed set of Y is both an IF-open set and an IF-closed (resp., IF-regular open) set of X. Proof.Obvious.Let f : ( , ) X  → ( , ) Y  be a mapping, and let g : X → X  Y be its graph.If g is IFPC (resp., IFCC) so f is IFPC (resp., IFCC).Proof.Let λ be an IF-open set of Y. Then 1  λ is an IF-open set of X  Y. Since g is IF-per- fectly continuous, g ← (1  λ) is both an IF-open set and an IF-closed set of X.Then we have

Definition 3 . 3 .
is both an IF-open set and an IF-closed set of X. Hence f is IFPC.The proof for IFCC is by the same fashion.□Let ( , )X  be an IF-ts, and let.IF-perfectly retract (IFPR, for short) (resp., IF-completely retract, IFR-retract) (resp., IFCR, IFRR, for short) of ( , ) X  if there exists an IFPC (resp., IFCC, IFRC) mapping r : ( , ) X  → ( , )A A  such that r(a)  a for all a .A  In this case r is called an IF-perfectly retraction (resp., IF-completely retraction, IF-R-retraction).

Remark 3 . 1 .Example 3 . 3 .
The implications between these different concepts are given by the following diagram:IFPR  IFCR.The converse of the above implication need not be true in general, as shown by the following examples.Let λ 1 and λ 2 be IF-sets on X  {a,b} defined by

Example 3 . 4 .
Let = { , , }, = { } X, X a b c A a  and let λ 1 and λ 2 be IF-sets on X defined by

Theorem 4 . 1 .
Let c I  be an interval-valued intuitionistic fuzzy compact space, let c nearly I  be an intervalvalued intuitionistic fuzzy nearly compact space, and let c almost I be an interval-valued intuitionistic fuzzy almost compact space.Then the implications between these different concepts are given by the following diagram:.


be interval-valued intuitionistic fuzzy-regular spaces.Let f : ( , )X  → ( , )Y  be a surjection and interval-valued intuitionistic fuzzy-almost continuous.If ( , ) X  is ,  be an o I  of Y. Then from the interval-valued intuitionistic fuzzy-almost continuity of f it follows that { from the interval-valued intuitionistic fuzzy-almost continuity of f, we see that interval-valued intuitionistic fuzzy-almost continuous containing (
is a family of interval-valued intuitionistic fuzzy regular-regular open sets, there exists a finite subfamily(1,..., )

□Theorem 4 . 6 .□Theorem 4 . 7 .□Definition 4 . 3 .
 (i) Obvious, since every interval-valued intuitionistic fuzzy regular-regular open cover is an interval-valued intuitionistic fuzzy regular-open cover.The image of an c nearly I  space under a mapping that is both interval-valued intuitionistic fuzzy regular-almost continuous and an interval-valued intuitionistic fuzzy regular-almost open surjection is The proof of this theorem follows a similar pattern as that of Theorem 4.2.The image of an c nearly I  space under an interval-valued intuitionistic fuzzy regular-strong continuity is an c I  space.Proof.The proof of this theorem follows a similar pattern as that of Theorem 4.2.An interval-valued intuitionistic fuzzy topological space ( , )X  is an interval-valued intuitionistic fuzzy S 1 -regular space iff for each   interval-valued intuitionistic fuzzy set X can be written as { IVIF-S 1 -regularity of ( , ), X  it follows that { ( ) : ( ) }.
the surjectivity and fuzzy strong continuity of f we