1

Chapter 1

Introduction

1.1 Fuzzy Sets and Intuitionistic Fuzzy Sets

Most of our traditional tools modeling, reasoning and computing are

crisp, deterministic and precise in character. By crisp we mean di-

chotomies, that is, yes-or-no type rather than more-or-less type. Cer-

tainty eventually indicates that we assume the structures and parameters

of the model to be definitely known and that there are no doubts about

their values or their occurrence.

For factual model or modeling languages two major complications arise:

(i) Real situations are very often not crisp and deterministic and they

cannot be described precisely.

(ii) The complete description of a real system often would require by

far more detailed data than a human being could ever recognize,

process and understand simultaneously.

In classical set theory, the membership of elements in a set is assessed

in binary terms according to a bivalent condition- an element either

belongs or does not belong to the set.

2

In 1923, the philosopher Russell [58] referred to the first point when

he wrote: ” All traditional logic habitually assumes that precise symbols

are being employed. It is therefore not applicable to this terrestrial life

but only to an imagined celestial existence”.

Zadeh [72] referred to the second point when he wrote: ”As the com-

plexity of a system increases, our ability to make precise and yet sig-

nificant statement about its behaviour diminishes until a threshold is

reached beyond which precision and significance (or relevance) become

almost mutually exclusive characteristics”.

Let us consider characteristic features of real-world systems again:

Real situations are very often uncertain or vague in a number of ways.

Due to lack of information the future state of the system might not be

known completely. This type of uncertainty (stochastic character) has

long been handled appropriately by probability theory and statistics.

This kolmogoroff type probability is essentially frequent and is based on

set-theoretic considerations.

Koopman’s probability refers to the truth of statement and therefore

it is based on logic. On both types of probabilistic approaches it is

assumed, however, that the events ( elements of sets ) or the statements,

respectively, are well defined.

We shall call this type of uncertainty or vagueness stochastic uncer-

tainty by contrast to the vagueness concerning the description of the

semantic meaning of the events, phenomena or statement themselves,

3

which we shall call fuzziness.

Fuzziness can be found in many areas of daily life, such as in Engineer-

ing (Blockley [18]), in Medicine (Vila and Delgado [69]), in Meteorology

( Cao and Chen [22] ), in Manufacturing ( Mamdani [48] ) and others.

It is particularly frequent, however, the meaning of a word might even

be well defined, but when using the word as a label for a set, the bound-

aries within which objects do or do not belong to the set become fuzzy

or vague.

Zadeh [72] writes: ”The notion of a fuzzy set provides a convenient

point of departure for the construction of a conceptual framework which

parallels in many respects the framework used in the case of ordinary

sets but is more general than the latter and, potentially, may prove to

have a much wider scope of applicability, particularly in the fields of

pattern classification and information processing.

Essentially, such a framework provides a natural way of dealing with

problems in which the source of imprecision in the absence of sharply

defined criteria of class membership rather than the presence of random

variables”.

Fuzzy set theory provides a strict mathematical framework in which

vague conceptual phenomena can be precisely and rigorously studied.

Fuzziness has so far not been defined uniquely semantically and probably

never will. It will mean different things, depending on the application

area and the way it is measured.

4

In the meantime, numerous authors have contributed to this theory.

Fuzzy sets are having useful and interesting applications in various fields

including Probability theory, Information theory [59] and Control [61].

As an extension of classical set theory, fuzzy set theory permits the

gradual assessment of the membership of elements in a set; this is de-

scribed with the aid of a membership function valued in the real unit

interval [0, 1] .

The theory of intuitionistic fuzzy sets further extends both the con-

cepts by allowing the assessment of the elements by two functions: for

membership and for non-membership, which belong to the real unit in-

terval [0, 1] and whose sum belongs to the same interval, as well.

Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator func-

tions of fuzzy sets are special cases of the membership and the non-

membership functions and of intuitionistic fuzzy sets, in the case when

the strict equality exists: i.e. the non-membership function fully com-

plements the membership function to 1, not leaving room for any uncer-

tainty.

Intuitionistic fuzzy sets can be more precisely expressed. For example,

the fact that the temperature of a patient changes, and other symptoms

are not quite clear. There is a fair chance of the existence of a non-null

hesitation part at each moment of evaluation of an unknown object.

Intuitionistic fuzzy sets as a generalization of fuzzy sets can be useful in

situations when description of a problem by a (fuzzy) linguistic variable,

5

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, financial services, etc.

1.2 Review Of Literature

The concept of a fuzzy set provides a natural framework for generaliz-

ing many of the concepts of a general topology. The theory of fuzzy

topological spaces was introduced and developed by Chang [24]. Since

then various notions in classical topology have been extended to fuzzy

topological spaces by fuzzy topologists like Azad [10], Zadeh [72], Tuna

Hatice Yalvac [67,68], Brown [20], Hutton And Reilly [40], Lowen [45,46],

Warren [70], Friedler [33], Gantner et. al. [34], Arya [3], Bin Shahna [16],

Ceck [23], Hewitt [39] and Necla Turanli [51]. Various departures from

the classical topology have also been observed in fuzzy topology. The

field of Mathematical Sciences which goes under the name of topology is

concerned with all questions directly or indirectly related to continuity.

Fuzzy topologists have introduced and investigated many different types

of fuzzy continuity.

After the introduction of the concept of fuzzy sets by Zadeh [72] several

researches were conducted on the generalizations of the notion of fuzzy

set. The concept of ”intuitionistic fuzzy set” was first published by

Atanassov [4] and many works by the same author and his colleagues

appeared in the literature [5,6,7,8]. Later topological structures in fuzzy

6

topological spaces [24] is generalized to ”intuitionistic fuzzy topological

spaces” by Coker in [30], and then the concept of ”intuitionistic set”

is introduced by Coker in [31]. Later this concept was generalized to

”intuitionistic L-fuzzy sets” by Atanassov And Stoeva [6]. This concept

is the discrete form of intuitionistic fuzzy set, and it is one of several ways

of introducing vagueness in mathematical objects. Levine [43], Mashhour

et. al.[49] and Njastad [52] introduced semi-open sets, preopen sets and

α -sets respectively. The concept of β -open sets in topological spaces

was introduced by Abd. El-Monsef et al [1]. The concept of intuitionistic

fuzzy g-closed sets was introduced by Thakur and Rekha Chaturvedi

[64]. Thangaraj and Balasubramanian [65] introduced and studied the

concepts of fuzzy basically disconnected spaces. The concepts of fuzzy

Hausdorff topological spaces was first introduced and studied by Rekha

Srivastava et.al.[54]. The concepts of fuzzy S- closed spaces was studied

by Coker [27].

Fuzzy topologists have introduced and investigated many different gen-

eralizations of fuzzy continuous functions. The concept of fuzzy Gδ set

was first introduced and studied by Balasubramanian [13]. The concepts

of connectedness and disconnectedness in fuzzy topological spaces have

been studied by Balasubramanian [12,14]. Roja, Uma and Balasubra-

manian [55,56,57] studied the concepts of continuity, connectedness and

disconnectedness by using fuzzy Gδ sets. The concept of semi θ con-

tinuity in an intuitionistic fuzzy topological spaces was introduced and

studied by Hanafy, Abd El-Aziz and Salman [38].

7

Lowen et. al.[47] introduced and developed the concept of fuzzy con-

vergence spaces. Geetha [35] established the concept of F-semigroup

compactifications. Sostak [60] introduced the concepts of L-fuzzy quasi

uniformity. He also studied several properties of L-fuzzy quasi uni-

form space and the topology generated by L-fuzzy quasi uniformity.

Cocker [28,29,30] introduced and studied various concepts like continuity,

compactness, connectedness and disconnectedness in intuitionistic fuzzy

topological spaces.The concept of fuzzy faintly continuous functions and

fuzzy faintly α -continuous functions are introduced and studied by An-

jan Mukherjee [9].

The concepts of bicompact extension of topological spaces was intro-

duced and developed by Aleksandrov [2]. The concepts of extensions of

topologies were introduced and developed by Borges [19] . Balasubrama-

nian [11] fuzzified the concept of extensions of topological spaces. Roja,

Uma and Balasubramanian [55] extended the concept of extensions of

fuzzy topological spaces to fuzzy bitopological spaces. The concept of

closure spaces was introduced and studied by Cech [23]. The notion of

fuzzy closure spaces was first introduced and developed by Mashhour

and Ghanim [50]. Chawalit Boonpok [26] introduced the concept of bi-

closure spaces. Tapi and Navalakhe [62,63] have introduced the concept

of fuzzy biclosure spaces. The generalisation of fuzzy continuous func-

tions was studied by Balasubramanian and Sundaram [14]. The concepts

of g-closed sets was introduced by Levine [44]. The extension principle

of closure operator is studied by Biacino [17].

8

1.3 Outline of the Thesis

This section presents a chapterwise summary of results obtained on in-

tuitionistic fuzzy regular Gδ sets, intuitionistic fuzzy regular Gδ contin-

uous functions, intuitionistic fuzzy regular semi (resp., pre, α and β )

open sets, intuitionistic fuzzy regular Gδ compact (resp., connected and

normal) spaces, intuitionistic fuzzy regular Gδ T1/2 spaces, intuitionis-

tic fuzzy convergence spaces, intuitionistic fuzzy convergence topologi-

cal spaces, intuitionistic fuzzy convergence topological semigroups, intu-

itionistic fuzzy Bohr lim compactifications, intuitionistic fuzzy lim semi

group compactifications, intuitionistic fuzzy quasi uniform topological

spaces, intuitionistic fuzzy quasi uniform regular Gδ sets, intuitionistic

fuzzy quasi uniform faintly regular Gδ continuous functions, intuition-

istic fuzzy quasi uniform regular Gδ compact spaces, simple extension

of intuitionistic fuzzy quasi uniform bitopological spaces, pairwise intu-

itionistic fuzzy quasi uniform connected spaces, pairwise intuitionistic

fuzzy quasi uniform disconnected spaces, pairwise intuitionistic fuzzy

quasi uniform regular Gδ basically disconnected spaces, pairwise intu-

itionistic fuzzy quasi uniform extremally disconnected spaces, pairwise

intuitionistic fuzzy quasi uniform Ti(i = 0, 1 and 2) spaces, intuitionistic

fuzzy closure spaces, intuitionistic fuzzy biclosure spaces, intuitionistic

fuzzy C -open (resp., closed) sets and intuitionistic fuzzy GC Hausdorff

(resp., normal and regular) biclosure spaces.

In chapter 2, the concept of intuitionistic fuzzy regular Gδ sets is

9

introduced. Some interesting properties are studied. Interesting interre-

lations are established. Counter examples are given wherever necessary.

Also the concept of intuitionistic fuzzy regular Gδ continuous functions

is introduced. Interesting properties are studied. The concept of intu-

itionistic fuzzy regular Gδ compact (resp., connected and normal) spaces

is introduced and studied.

In chapter 3, the concept of intuitionistic fuzzy convergence topologi-

cal spaces is introduced by using the limit function (in short, lim). The

concept of intuitionistic fuzzy convergence topological semigroups is in-

troduced and studied. Besides by providing several propositions, intu-

itionistic fuzzy Bohr lim compactifications and intuitionistic fuzzy lim

semigroup compactifications are established.

The main motive of chapter 4 is to introduce the concept of intuitionis-

tic fuzzy quasi uniform topological spaces and to discuss the interesting

properties of intuitionistic fuzzy quasi uniform faintly regular Gδ con-

tinuous functions. Some interesting interrelations are established with

suitable examples. Further, the concept of intuitionistic fuzzy quasi

uniform regular Gδ compact spaces is introduced. In this connection,

interrelations are studied.

Chapter 5 deals with the extension of intuitionistic fuzzy quasi uni-

form bitopological spaces. Some interesting characterization of pair-

wise intuitionistic fuzzy quasi uniform connected spaces are established.

Properties of pairwise intuitionistic fuzzy quasi uniform compact (resp.,

10

Lindelof ) spaces are studied. In this connection, separation axioms are

established.

In chapter 6, the concept of intuitionistic fuzzy closure spaces is in-

troduced. Some interesting properties are studied. The concept of in-

tuitionistic fuzzy biclosure spaces is introduced. Several properties are

established. The concept of intuitionistic fuzzy GC Hausdorff (resp.,

normal and regular) spaces are introduced and studied.

1.4 Basic Concepts

In this scetion, some basic concepts of topology, intuitionistic topology,

fuzzy topology and intuitionistic fuzzy topology have been recalled. Also,

related results, propositions and important theorems are collected from

various research articles.

Throughout this thesis X be a non empty set, I = [0, 1] and ζX is

the set of all intuitionistic fuzzy sets.

Definition 1.4.1. [24]

Let X be a non-empty set and I be the unit interval. A fuzzy set in X

is an element of the set I X of all functions from X to I.

Definition 1.4.2. [24]

Let (X, T ) be any fuzzy topological space. The characteristic function

11

of a subset A of X is denoted by χA and defined as

χA(x) =

0 if x ƒ∈ A

Definition 1.4.3. [4]

1 if x ƒ∈ A

Let X be a non empty fixed set and I the closed interval [ 0, 1 ].

An intuitionistic fuzzy set ( IFS ) A is an object of the following form

A = {(x, µA(x), γA(x)) : x ∈ X } where the functions µA : X → I

and γA : X → I denote the degree of membership ( namely µA(x) )

and the degree of non membership ( namely γA(x) ) for each element

x ∈ X to the set A respectively and 0 ≤ µA(x) + γA(x) ≤ 1 for each

x ∈ X. Obviously, every fuzzy set A on a nonempty set X is an IFS

of the following form A = {(x, µA(x), 1 − µA(x)) : x ∈ X } . For the

sake of simplicity, we shall use the symbol A = (x, µA(x), γA(x)) for the

intuitionistic fuzzy set A = {(x, µA(x), γA(x) : x ∈ X )} .

For a given non empty set X, denote the family of all intuitionistic

fuzzy sets in X by the symbol ζX .

Definition 1.4.4. [4]

Let A and B be the intuitionistic fuzzy sets of the form A = {(x, µA(x),

γA(x)) : x ∈ X } and B = {(x, µB (x), γB (x)) : x ∈ X } . Then

(i) A ⊆ B if and only if µA(x) ≤ µB (x) and γA(x) ≥ γB (x).

(ii) A = {(x, γA(x), µA(x)) : x ∈ X }.

(iii) A ∩ B = {(x, µA(x) ∧ µB (x), γA(x) ∨ γB (x)) : x ∈ X }.

12

(iv) A ∪ B = {(x, µA(x) ∨ µB (x), γA(x) ∧ γB (x)) : x ∈ X }.

(v) A = B if and only if A ⊆ B and B ⊆ A.

(vi) [ ]A = {(x, µA(x), 1 − µA(x)) : x ∈ X }.

(vii) ( )A = {(x, 1 − γA(x), γA(x)) : x ∈ X }.

Definition 1.4.5. [30]

The intuitionistic fuzzy sets 0∼ and 1∼ are defined by 0∼ = {(x, 0, 1) :

x ∈ X } and 1∼ = {(x, 1, 0) : x ∈ X }

Definition 1.4.6. [30]

Let {Ai/i ∈ J } be an arbitrary family of intuitionistic fuzzy sets in

X. Then

(i) T

Ai = {(x, V

µA(x), W

γA(x)) : x ∈ X }.

(ii) S

Ai = {(x, W

µA(x), V

γA(x)) : x ∈ X }.

Corollary 1.4.1. [30]

Let A, B and C be intuitionistic fuzzy sets in X. Then

(i) A ⊆ B and C ⊆ D ⇒ A ∪ C ⊆ B ∪ D and A ∩ C ⊆ B ∩ D.

(ii) A ⊆ B and A ⊆ C ⇒ A ⊆ B ∩ C.

(iii) A ⊆ C and B ⊆ C ⇒ A ∪ B ⊆ C.

(iv) A ⊆ B and B ⊆ C ⇒ A ∪ B ⊆ C.

(v) A ⊆ B and B ⊆ C ⇒ A ⊆ C.

13

1

(vi) A ∪ B = A ∩ B.

(vii) A ∩ B = A ∪ B.

(viii) A ⊆ B ⇒ B ⊆ A.

(ix) A = A.

(x) 1∼ = 0∼. (xi) 0∼

= 1∼. Definition

1.4.7. [30]

Let X and Y be two non empty sets and f : X → Y be a function.

(i) If B = {(y, µB (y), γB (y)) : y ∈ Y } is an intuitionistic fuzzy set

in Y, then the inverse image of B under f is an IFS defined by

f −1(B) = {(x, f −1(µB )(x), f −1(γB )(x)) : x ∈ X } .

(ii) If A = {(x, λA(x), ϑA(x)) : x ∈ X } is an intuitionistic fuzzy set

in X. Then the image of A under f is an IFS defined by f (A) =

{(y, f (λA)(y), (1 − f (1 − ϑA))(y)) : y ∈ Y } where

f (λA)(y) =

supx∈f −1 (y)λA(x) f −

(y) = 0

0 otherwise

(1 − f (1 − ϑA))(y) =

1 infx∈f −1 (y)ϑA(y) f −

(y) = 0

Definition 1.4.8. [30]

0 otherwise

Let A, Ai (i ∈ J ) be IFSs in X. B, Bj (j ∈ K) IFSs in Y and

f : X → Y be a function. Then

14

∼

∼

(i) A1 ⊆ A2 ⇒ f (A1) ⊆ f (A2).

(ii) B1 ⊆ B2 ⇒ f −1(B1) ⊆ f −1(B2).

(iii) A ⊆ f −1(f (A)) {If f is injective, then A = f −1(f (A))}.

(iv) f (f −1(B)) ⊆ B {If f is surjective, then A = f (f −1(A))}.

(v) f −1(S

Bj ) = S

f −1(Bj ).

(vi) f −1(T

Bj ) = T

f −1(Bj ).

(vii) f (S

Aj ) = S

f (Aj ).

(viii) f (T

Aj ) ⊆ T

f (Aj ). {If f is injective, then f (T

Aj ) = T

f (Aj )}.

(ix) f −1(1 ) = 1∼.

(x) f −1(0 ) = 0∼.

(xi) f (1∼) = 1∼, if f is surjective.

(xii) f (0∼) = 0∼.

(xiii) f (A) ⊆ f (A), if f is surjective.

(xiv) f −1(B) ⊆ f −1(B).

Definition 1.4.9. [30]

An intuitionistic fuzzy topology ( IFT ) in Coker’s sense on a non

empty set X is a family τ of IFSs in X satisfying the following axioms.

(i) 0∼ , 1∼ ∈ τ.

15

(ii) G1 ∩ G2 ∈ τ for any G1, G2 ∈ τ.

(iii) ∪Gi ∈ τ for arbitrary family {Gi/i ∈ I } ⊆ τ.

In this case the pair (X, τ ) is called an intuitionistic fuzzy topological

space (IFTS for short) and any IFS in τ is known as an intuitionistic

fuzzy open set (IFOS for short) in X. The complement A of an IFOS A

in X is called an intuitionistic fuzzy closed set (IFCS for short) in X.

Definition 1.4.10. [30]

Let (X, τ ) be an IFTS and A = (x, µA, γA) be an IFS in X. Then the

fuzzy interior and fuzzy closure of A are defined by

int A = ∪{G / G is an IFOS in X and G ⊆ A}.

cl A = ∩{G / G is an IFCS in X and G ⊇ A}.

Proposition 1.4.1. [30]

For any IFS A in ( X , τ ) we have

(i) cl(A) = int(A). (ii)

int(A) = cl(A).

Definition 1.4.11. [30]

Let (X, τ ) be an IFTS. cl(A) is an IFCS and int(A) is an IFOS in

X. Then

(i) A is an IFCS in X iff cl(A) = A.

16

(ii) A is an IFOS in X iff int(A) = A.

Proposition 1.4.2. [30]

Let (X, τ ) be an IFTS and A, B be IFSs in X. Then the following

properties hold.

(i) int(A) ⊆ A.

(ii) A ⊆ cl(A).

(iii) A ⊆ B ⇒ int(A) ⊆ int(B).

(iv) A ⊆ B ⇒ cl(A) ⊆ cl(B).

(v) int(int(A)) = int(A).

(vi) cl(cl(A)) = cl(A).

(vii) int(A ∩ B) = int(A) ∩ int(B).

(viii) cl(A ∪ B) = cl(A) ∪ cl(B).

(ix) int(1∼) = 1∼. (x)

cl(0∼) = 0∼. Definition

1.4.12. [30]

A function f : X → Y from an IFTS X into an IFTS Y is called an

intuitionistic fuzzy continuous if f −1(B) is an IFOS in X , for each IFOS

in Y.

17

Definition 1.4.13. [30]

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 iff the image of each IFS in τ is an IFS

in Φ.

Definition 1.4.14. [9]

Let f : (X, T ) → (Y, S) be a function. The graph g : X → X × Y of

f is defined by g(x) = (x, f (x)) ∀x ∈ X.

Definition 1.4.15. [37]

Let X, Y be non empty sets and U = {(x, λU (x), ϑU (x)) : x ∈ X },

V = {(y, λV (y), ϑV (y)) : y ∈ Y } IFSs of X and Y respectively. Then

U × V is an IFS of X × Y defined by:

(U × V ) = ((x, y), min(µU (x), µV (y)), max(γU (x), γV (y))).

Definition 1.4.16. [36,42]

Let A be an IFS of an IFTS X. Then A is called an

(i) intuitionistic fuzzy regular open set (IFROS) if A = int(cl(A)).

(ii) intuitionistic fuzzy semi open set (IFSOS) if A ⊆ cl(int(A)).

(iii) intuitionistic fuzzy β open set (IF β OS) if A ⊆ cl(int(cl(A))).

(iv) intuitionistic fuzzy preopen set (IFPOS) if A ⊆ int(cl(A)).

(v) intuitionistic fuzzy α open set (IF α OS) if A ⊆ int(cl(int(A))).

18

i=1

Definition 1.4.17. [36,42]

Let A be an IFS in IFTS.Then

(i) βint(A) = ∪{G / G is an IF β OS in X and G ⊆ A} is called an

intuitionistic fuzzy β -interior of A.

(ii) βcl(A) = ∩{G / G is an IF β CS in X and G ⊇ A} is called an

intuitionistic fuzzy β -closure of A.

(iii) ints (A) = ∪{G / G is an IFSOS in X and G ⊆ A} is called an

intuitionistic fuzzy semi interior of A.

(iv) cls (A) = ∩{G / G is an IFCS in X and G ⊇ A} is called an

intuitionistic fuzzy semi closure of A.

Definition 1.4.18. [56]

A fuzzy topological space ( X, T ) is said to be fuzzy β − T1/2 space

if every gf β - closed set in ( X,T) is fuzzy closd in (X, T ) .

Definition 1.4.19. [13]

Let ( X, T ) be a fuzzy topological space and λ be a fuzzy set in X.

λ is called fuzzy Gδ set if λ = Vi=∞ λi where each λi ∈ T .

The complement of a fuzzy Gδ is a fuzzy Fσ set.

Definition 1.4.20. [38]

A function f : (X, Ψ) → (Y, Φ) is said to be intuitionistic fuzzy semi

θ - continuous if for each IFP c(a, b) in X and V ∈ N sq (f (c(a, b))) , there

exists U ∈ N θq (c(a, b)) such that f (U ) ≤ V.

19

Definition 1.4.21. [66]

Let S be a non empty set and ◦ be a binary operation on S . The

algebraic system (S, ◦) is said to be a semigroup if the operation ◦ is

associative.

Definition 1.4.22. [66]

Let (S, ∗) be a semigroup and T ⊆ S . If the set T is closed under

the operation ∗ , then (T , ∗) is said to be a subsemigroup of (S, ∗) .

Definition 1.4.23. [66]

Let (S, ◦) and (Y, ∗) be two algebraic systems of the same type in the

sense that both ◦ and x are binary operations. A function g : X → Y is

said to be a homomorphism from (S, ◦) and (Y, ∗) if for any x1, x2 ∈ X ,

g(x1 ◦ x2) = g(x1) x g(x2) .

Definition 1.4.24. [66]

Let g be a homomorphism from (X, ◦) to (Y, ∗) . If g : X → Y is

one to one and onto then g is said to be an isomorphism.

Definition 1.4.25. [66]

A semigroup is a non empty set S together with an associative multi-

plication (x, y) → xy from S × S into S . If S has a Hausdorff topology

such that (x, y) → xy is continuous with the product topology on S ×S ,

then S is said to be topological semigroup.

20

Definition 1.4.26. [47]

Given a set X, the pair (X, lim) is said to be a fuzzy convergence

space, where lim : F(X ) → I X , provided:

(i) limF = infG∈Pm (F)limG, F ∈ F(X ).

(ii) limF ≤ c(F), F ∈ FP (X ).

(iii) limG ≤ limF when F ⊆ G, F, G ∈ FP (X ).

(iv) limα1̇ x ≥ α1x for 0 < α ≤ 1 and x ∈ X .

Definition 1.4.27. [71]

Let (X, F ) be a fuzzy topological space, xλ, 0 < λ ≤ 1, be a fuzzy

point and P a closed fuzzy set in X. Then P is called a remoted neigh-

bourhood (R-nbd) of xλ if x ƒ∈ P.

Definition 1.4.28. [71]

A fuzzy net S is a function S : D → T where D is a directed set with

order relation ≤ and T the collection of all fuzzy points in X. Then

for each n ∈ D, S(n) is a fuzzy point belonging to T . Let its value in

(0, 1] be denoted by λn. Thus we get a crisp net V (S) = {λn : n ∈ D}

in the half open interval (0, 1]. If V (S) converges to α ∈ (0, 1] we say

V (S) is an α− net.

Definition 1.4.29. [35]

Let X be a semigroup and F a T2 fuzzy topology on X. Then (X, F )

is called a fuzzy topological semi group (FTSG) if for all x, y ∈ X,

21

the function α : (x, y) → xy of (X, F ) × (X, F ) into (X, F ) is fuzzy

continuous.

Definition 1.4.30. [35]

Let X be a FTSG. The Bohr fuzzy compactification of X is a pair

(α, Y ) such that Y is a compact FTSG. α : X → Y is a F- morphism

provided whenever β : X → Z is an F-morphism of X into a compact

FTSG Z, then there exist a unique F-morphism f : Y → Z such that

f ◦ α = β.

Definition 1.4.31. [35]

Let X denote an FTSG. By an F-semigroup compactification of X we

mean an ordered pair (α, A) , where A is a compact FTSG and α : X →

A is a dense F-marphism.

Definition 1.4.32. [53]

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

(a1, b1) , (a2, b2) be any two intuitionistic fuzzy pairs. Then define

(i) (a1, b1) ≤ (a2, b2) if and only if a1 ≤ a2 and b1 ≥ b2 .

(ii) (a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2 .

(iii) If {(ai, bi/i ∈ J )} is a family of intuitionistic fuzzy pairs, then

∨(ai, bi) = (∨ai, ∧bi) and ∧(ai, bi) = (∧ai, ∨bi) .

(iv) The complement of an intuitionistic fuzzy pair (a, b) is the intu-

itionistic fuzzy pair defined by (a, b). = (b, a)

22

(v) 1∼ = (1, 0) and 0∼ = (0, 1) .

Definition 1.4.33. [12]

A fuzzy topological space X is said to be fuzzy extremally disconnected

if the closure of every fuzzy open set in X is fuzzy open in X.

Definition 1.4.34. [28]

(i) Let X be a non empty set. If r ∈ I0, s ∈ I1 are fixed real number,

such that r + s ≤ 1 then the intuitionistic fuzzy set xr,s is called

an intuitionistic fuzzy point (IFP for short) in X given by

xr,s(xp) =

(r, s) if x = xp

(0, 1) if x = xp

for xp ∈ X is called the support of xr,s where r denotes the degree

of membership value and s is the degree of non membership value

of xr,s

(ii) An intuitionistic fuzzy point xr,s is said to belong to an intuitionistic

fuzzy set A if r ≤ µA(x) and s ≥ γA(x).

(iii) An intuitionistic fuzzy singleton xr,s ∈ X is an intuitionistic fuzzy

set in X taking value r ∈ (0, 1] and s ∈ [0, 1) at x and 0∼ else-

where.

(iv) Two intuitionistic fuzzy point or intuitionistic fuzzy singleton are

said to be distinct if their support are distinct.

23

Definition 1.4.35. [71]

Let (X, F ) be a fuzzy topological space. A fuzzy set A in X is called

N-compact if each α− net, 0 < α ≤ 1, contained in A has at least a

cluster point xα with value α. When A is equal to the constant fuzzy set

1 and is N-compact, we call (X, F ) an N − compact fuzzy topological

group.

Definition 1.4.36. [60]

Let L be a completely distributive lattice and X be a set. Let D denote

the family of function U : LX → LX such that

(i) M ⊆ U (M ), for each M ∈ LX .

(ii) U (∨Mj ) = ∨j U (Mj ) for each family {Mj /j ∈ J } ⊂ LX .

A L-fuzzy quasi uniformity on X can now be defined as a subset U ⊆ D

satisfying the following axioms.

(i) If U ∈ U , U ⊆ V and U ∈ D, then V ∈ U .

(ii) If U1, U2 ∈ U , then there exists V ∈ U such that V ≤ U1 ∧ U2.

(iii) For every U ∈ U there exists V ∈ U such that V ◦ V ≤ U.

A L-fuzzy quasi uniformity is called an L-fuzzy uniformity, if U ∈ U

implies U −1 ∈ U (U −1 : LX → LX is defined by U −1(N ) = ∧{M/U (M c) ≤

N c}). The pair (X, U ) is called a Hutton L-fuzzy (quasi) uniform space.

24

Definition 1.4.37. [60]

Every L-fuzzy quasi uniformity generates an L-fuzzy topology. Let

(X, U ) be an L-fuzzy quasi uniform space. Then the operator I nt :

LX → LX defined by I nt(M ) = ∨{N ∈ LX /U (N ) ≤ M for some U ∈

U } is the fuzzy interior operator. The corresponding L-fuzzy topology

X

on X, τU = {M ∈ L

U .

/I nt(M ) = M } is called generated topology by

Definition 1.4.38. [28]

Let A and B be intuitionistic fuzzy set in an intuitionistic fuzzy topo-

logical space (X, T ) . Then A is said to be quassi coincident with B

denoted by AqB if there exists an x ∈ X such that γB (x) < µA(x) or

µA(x) > γB (x) .

Definition 1.4.39. [28]

An intuitionistic fuzzy point xr,s in X is said to be quassi coincident

with an intuitionistic fuzzy set A, denoted by xr,sqA if and only if r >

γA(x) or s < µA(x) .

Definition 1.4.40. [21]

A fuzzy point xr in an fts X is said to be fuzzy fc- accumulation point

of a fuzzy filterbase B if for each fuzzy feebly open feeble quassi-nbd U

of xr and for each B ∈ B , BqF cl(U ).

Definition 1.4.41. [30]

Let ( X, T ) be an intuitionistic fuzzy topological space .Then ( X, T ) is

25

said to be an intuitionistic fuzzy compact space if for every intuitionistic

fuzzy open cover {Vi/i ∈ J } of ( X, T ) there exists a finite subset J0

of J such that S{Vi/i ∈ J0} = 1∼.

Definition 1.4.42. [29]

Let (X, T ) be an intuitionistic fuzzy topological space. Then (X, T )

is said to be an intuitionistic fuzzy almost compact space if for every

intuitionistic fuzzy open cover {Vi/i ∈ J } of ( X, T ) there exists a

finite subset J0 of J such that S{I F cl(Vi)/i ∈ J0} = 1∼.

Definition 1.4.43. [30]

Let (X, T ) be an intuitionistic fuzzy topological space. Then X is said

to be a fuzzy

(i) C5 - disconnected space if there exists an intuitionistic fuzzy open

set and intuitionistic fuzzy open set G such that G = 1∼ and G =

0∼.

(ii) C5 - connected space if it is not an intuitionistic fuzzy C5 - discon-

nected space.

Definition 1.4.44. [15]

A fuzzy topological space (X, τ1, τ2) is said to be pairwise fuzzy basi-

cally disconnected space if τ1 - closure of each τ2 fuzzy open, τ2 fuzzy

Fσ is τ2 fuzzy open and τ2 - closure of each τ1 fuzzy open, τ1 fuzzy Fσ

is τ1 fuzzy open.

26

Definition 1.4.45. [25]

A fuzzy topological space (X, τ1, τ2) is said to be pairwise fuzzy ex-

tremally disconnected space τ1 - closure of each τ2 fuzzy open is τ2 fuzzy

open and τ2 - closure of each τ1 fuzzy open is τ1 fuzzy open.

Definition 1.4.46. [55]

Let (X, τ1, τ2) be a fuzzy bitopological space and let δ ƒ∈ τ1 and δ ƒ∈

τ2. If τ1(δ) and τ2(δ) are simple extensions of the fuzzy topologies τ1

and τ2 respectively, then the fuzzy bitopology (τ1(δ), τ2(δ)) is called a

simple extension of the fuzzy bitopology (τ1, τ2).

Definition 1.4.47. [66]

Let (P, ≤) be a partially ordered set and A ⊆ P. Any element x ∈ P

is an upper bound for A if for all a ∈ A, a ≤ x. Similarly, any element

x ∈ P is a lower bound for A if for all a ∈ A, x ≤ a.

Definition 1.4.48. [55]

Let (X, τ1, τ2) be a fuzzy bitopological space. (X, τ1, τ2) is said to

be pairwise fuzzy normal and if for every pair of fuzzy sets λ ∈ τ1 and

µ ƒ∈ τ2 such that λ ≤ µ, there exists τ1 - fuzzy open set δ or τ2 - fuzzy

open set δ such that λ ≤ δ ≤ clτi(δ) ≤ µ, for i = 1, 2.

Definition 1.4.49. [50]

A function u : I X → I X defined on the family I X of all fuzzy sets of

X is called a fuzzy closure operator on X and the pair (X, u) is called

fuzzy closure space, if the following conditions are satisfied.