CHAPTER 2 LITERATURE SURVEY -...

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CHAPTER 2

LITERATURE SURVEY

2.1 INTRODUCTION

Topological ideas are present in almost all the areas of today’s

mathematics. The subset of topology itself consists of several different

branches such as Point set Topology, Algebraic Topology and Differential

Topology which have relatively little in common. Many researchers

developed the new area in topology equipped with two topological branches

called Bitopology. The monograph is the first and the initial introduction to

the theory of bitopological spaces and its application. In particular, different

families of subsets of bitopological spaces are introduced and the

relationships between the two topologies are analyzed on one and the same

set. The theory of dimension of Bitopological spaces and the theory of Baire

bitopological spaces are constructed and the various classes of mappings of

bitopological spaces are studied. Fuzzy Mathematics forms a branch of

Mathematics related to Fuzzy Set Theory and Fuzzy Logic. Fuzzy Logic is a

form of many valued logic or probabilistic logic which deals with reasoning

that is approximate rather than fixed or exact reasoning. In contrast with the

traditional logical theory where binary sets have two valued logics true or

false, fuzzy logic variables may have true values that range in degree between

0 and 1.

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2.2 CLOSED SETS AND OPEN SETS IN TOPOLOGICAL

SPACES

Mashour et al. (1982), Levine (1963,1970), Njastad (1965), Abd El

-Monsef et al. (2005), Arya and Noiri (1990), Bhattacharya and Lahiri (1987),

Palaniappan and Rao (1993), Maki (1986), Maki et al. (1991) and Dontchev

(1995) introduced pre-open sets, semi-open sets , g-open sets, -open sets, -

open sets, generalized semi-open sets, semi-generalized open sets,

generalized^-sets, regular generalized open sets, generalized pre-open sets and

generalized semi-pre open sets respectively, which are some stronger and

weaker forms of open sets. From the above said concepts, the following

important and relevant results are obtained.

(1) Semi-open sets and semi-continuity in topological spaces:

Levine (1963).

The notion of semi-generalized closed set was introduced and

the properties were explored.

(2) Generalized closed set in topology:

Levine (1970).

The author introduced g-closed set. The standard properties

investigated in g- closed sets are normal space, complete

uniform space and locally compact Haustroff space. Also, T1

- space implies T1/2-space.

(3) Remarks on semi-generalized closed sets and generalized semi

-closed sets:

Maki et al. (1996).

The authors studied the product of generalized semi-closed

sets and proved a product theorem in GSO-Compact spaces.

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A new class of topological spaces that is Tgs-space was

introduced. The homeomorphic image of Tgs-space and the

relationships among the T1/2 -spaces, semi-T1/2-spaces and Tgs

-spaces were investigated.

(4) On e-open sets, Dp*-sets and Dpq*-sets and the

decomposition of continuity:

Erdal Ekici (2008).

The author introduced the idea of the notion of e-open sets

which is a generalization of -semi-open sets and -pre-open

sets.

(5) On generalized b-closed sets:

Ahmad Al-omari and Mohd.Salmi Md.Noorani (2009).

The authors proved, A be a gb-closed subset of (X, ). The

bcl(A)-A, then does not contain any non-empty closed sets. A

subset A X is gb-open, if and only if F bInt(A), whenever

F is a closed set and F A.

2.3 CONTINUOUS MAPS IN TOPOLOGICAL SPACES

Strong and Weak forms of continuous maps have been introduced

and investigated by several researchers like (Arockiarani(1997), Balachandran

et al. (1991), Dontchev (1996), Ganster and Reilly (1989), Levine (1960,

1961), Maheshwari and Thakur (1980,1985), Mashour et al. (1982, 1983) and

Narsef (2001)). The strong forms of continuous maps have been discussed by

Noiri (1974). Levine (1960), Arya and Gupta (1974) and Reilly and

Vamanamurthy (1983). They have introduced strong continuous maps,

strongly -continuous map, super continuous map and clopen continuous

maps respectively. Biswas (1969), Ganster and Reilly (1989), Noiri (1974),

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Mashour et al. (1982), Tong (1986) and Devi et al. (1993,1998) have

introduced and studied simple continuity, almost continuity, weak continuity,

-continuity, -continuity, semi-weak continuity, weak almost continuity,

-generalized continuity and generalized- -continuity respectively. Here, a

brief survey of some of the research papers published on the strong and weak

form of continuous maps is given.

(1) A note on semi-continuous mappings:

Noiri (1973).

The author has given the characterization of semi continuous

mappings and investigated some properties of such mappings.

He also investigated the properties of Hausdraf spaces and

semi continuous mappings.

(2) Semi-Generalized continuous maps and semi-T1/2-spaces:

Sundaram et al. (1991).

The authors have introduced the concept of semi-generalized

continuous maps. Moreover, they investigated the

semi-homeomorphic image of semi-T1/2-spaces and obtained a

product theorem for semi-T1/2-spaces.



The authors have given the characterization of various forms

of continuity that is associated with generalized b-closed

sets. They characterized the Tgs-spaces in terms of these

notion of continuity.

The authors defined a map f : X Y is said to be

approximately b-open (briefly ap-b-open) if bcl(F) f(U),

whenever U is an open subset of X, F is a gb-closed subset of

Y and F f(U).

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2.4 IRRESOLUTE MAPS IN TOPOLOGICAL SPACES

Crossley and Hildebrand (1972) introduced and investigated

irresolute maps which are stronger than the semi-continuous maps.

Cammaroto and Noiri (1989), Ganster and Reilly (1989), Maheshwari and

Thakur (1985), Dontchev (1996), Balachandran et al. (1996), Devi et al.

(1998) and Arokiarani (1997) introduced and studied quasi-irresolute and

strongly irresolute maps, -irresolute maps, weak irresolute maps and -

irresolute maps, gc-irresolute maps, g-irresolute maps, g -irresolute maps

and gr-irresolute maps respectively. A brief survey of some of the research on

strong and weak forms of irresolute maps that has been published is given

below.

(1) On -irresolute mappings:

Maheshwari and Thakur (1980).

The authors have introduced and investigated the properties of

-irresolute maps.

(2) Almost irresolute functions:

Cammaroto and Noiri (1989).

The authors have introduced and studied the notion of almost

irresolute function in topological spaces.



The authors proved that if the bijective f : X Y is b-irresolute

and open, f is then gb-irresolute.

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2.5 OPEN MAPS AND CLOSED MAPS IN TOPOLOGICAL

SPACES

Pre-closed mappings were introduced and studied by Sen and

Bhattacharya (1993). Noiri (1973,1984), Mashour et al. (1982,1983), Crossley

et al. (1972), Devi et al. (1993,1998), Noiri et al. (1998), Arochiarani (1997).

They defined and studied semi-open maps and semi-closed maps, semi-

generalized closed maps, generalized semi-closed maps, -open maps, pre

open maps, weak pre-open maps, -open maps, -closed maps, pre-semi open

maps, g-closed maps, g-closed maps and rg-closed maps respectively.

Some of the studies on open maps and closed maps are given below.

(1) Remarks on semi open mappings:

Noiri (1973).

The author has studied some of the properties of semi-open

mappings. Also, he studied the semi-open sets in subspaces

and semi-open sets in product spaces.

(2) A generalization of closed mappings:

Noiri (1973).

The author has introduced a class of mappings called

semi-closed mappings which contain the class of closed

mappings and gave a few characterizations of such

mappings.

2.6 HOMEOMORPHISM IN TOPOLOGICAL SPACES

Many researchers studied the topic of homeomorphism in topology.

Semi-homeomorphism which is weaker than a homeomorphism has been

discussed by Biswas (1969) and Crossly and Hilderbrand (1972).

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Semi-generalized homeomorphism and generalized semi-homeomorphism

-homeomorphism, g-homeomorphism and gc-homeomorphism have been

defined by Devi et al. (1995), Maki et al. (1991,1994,1996) and Arokiarani

(1997). Here the studies on generalized homeomorphism are discussed below.

(1) On Generalized homeomorphism in topological space:

Maki et al. (1991).

The authors have defined the notion of generalized-

homeomorphism and gc-homeomorphism which are

generalizations of homeomorphism and they have investigated

some of the properties of generalized-homeomorphism.

(2) Semi-generalized homeomorphism and generalized semi-

homeomorphism in topological spaces:

Devi et al. (1995).

The authors have introduced two classes of mappings namely

generalized semi-homeomorphism and semi-generalized

homeomorphism and investigated the group structure of their

sub-group.

2.7 BITOPOLOGICAL SPACES

Arya and Noiri (1990) studied the properties of quasi-semi-

components in bitopological spaces. Fukutake (1989), Khedr and Noiri (2007)

and Njastad (1965) have introduced semi-open sets, almost-s-continuous

function and on some classes of regularly open sets respectively. Mukherjee,

Banerjee and Malakar (1990) have introduced QHC-spaces in bitopological

spaces. Nagaveni (1999) has studied wg-closed sets in bitopological spaces.

The following are some of the references on the bitopology.

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(1) Weakly quasi-continuous function:

Popa and Noiri (2004).

The authors have introduced a novel notion of weakly quasi

-continuous functions. They obtained some of the

characterizations and properties of these functions.

(2) Separation axioms and Strongly Generalized Closed sets in

bitopological spaces.

Zakari and Al-Saadi (2008).

The authors have introduced strongly generalized closed sets

in bitopological spaces. Also, they discussed the

continuous, irresolute homeomorphism and their properties. A

bitopological space is pairwise-T1/2 space, if and only if {x} is

j-open or j -closed for each x X.

(3) On pairwise semi-generalized closed sets:

Fathi H.Khedr and Hanan S.Al-Saadi (2009).

The authors have introduced the class of semi-generalized

closed sets, which are properly placed between the classes of

generalized semi-closed sets and semi-closed sets. The authors

proved that if ji -sint(A) B A and A is ij-sg-open, then B is

ij-sg-open.

2.8 FUZZY TOPOLOGICAL SPACES.

Zadeh (1965) introduced the concept of fuzzy sets. Using fuzzy sets

Chang (1968) introduced fuzzy topological spaces. Several authors like Azad

(1981), Balasubramanian et al. (1991), and Wong (1973) have contributed to

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the development of fuzzy topology. Mukherjee and Sinha (1989) introduced

and investigated irresolute and almost open functions between fuzzy

topological spaces. Balasubramanian and Sundaram (1991) defined and

studied generalized closed sets and generalized continuous maps in fuzzy

topological spaces. The list below briefs some of the works done on fuzzy

topological space.

(1) Fuzzy topological product and quotient theorem :

Wong (1973).

The author has introduced and studied the properties of

product fuzzy topology and quotient- fuzzy topology.

(2) On fuzzy semi-continuity, fuzzy almost continuity and fuzzy

weakly continuity:

Azad (1981).

The author has introduced and studied the properties of fuzzy

semi-continuous, fuzzy almost continuous and fuzzy weakly

continuous maps in fuzzy topological space.

2.9 PRELIMINARIES

The basic definitions and results used for the present study is

outlined in this section.

2.9.1 Closed Sets and Open Sets

Definition 2.9.1.1: A subset A of (X, ) is called -open (Njastad

1965), if A int (cl(int(A)).

Definition 2.9.1.2: A subset A of (X, ) is called semi-open

(Levine 1963), if A cl(int (A)).

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Definition 2.9.1.3: A subset A of (X, ) is called pre-open

(Mashour 1982), if A int (cl(A)).

Definition 2.9.1.4: A subset A of (X, ) is called semi-pre-open

(Levine 1963), if A cl(int (cl(A))).

Definition 2.9.1.5: A subset A of a space X is said to be b-open

(Andrijevic 1996), if A cl (int (A)) int (cl(A)).

Definition 2.9.1.6: A generalized closed set (briefly g-closed)

(Levine 1970), if cl(A) U whenever A U and U is open.

Definition 2.9.1.7: An generalized-closed set (briefly g-closed)

(Mahi et al. 1994), if cl(A) U whenever A U and U is open.

Definition 2.9.1.8: A generalized pre-closed set (briefly gp-closed)

(Mahi et al. 1996), if A* U whenever A U and U is open.

Definition 2.9.1.9: A generalized semi-preclosed (briefly gsp

-closed) set (Dontchev 1995), if spcl(A) U whenever A U and U is open.

Definition 2.9.1.10: A generalized semi-closed set (briefly gs

-closed) (Devi et al. 1993) set, if scl(A) U whenever A U and U is open.

Definition 2.9.1.11: A semi generalized closed set (briefly sg

-closed) (Bhattacharya et al. 1987), if scl(A) U whenever A U and U is

semi open.

Definition 2.9.1.12: A generalized b-closed set (briefly gb-closed)

(Ahmad Al-omari and Mohd-Salmi Md.Noorani 2009), if bcl(A) U

whenever A U and U is open.

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Definition 2.9.1.13: A subset A of a topological space X, is called

g* - closed (Maragathavalli and Sheik John 2005 ), if cl(A) U whenever

A U and U is g-open in (X, ).

Definition 2.9.1.14: A subset A of a topological space X is called

s g*-closed set (Maragathavalli and Sheik John 2005), if cl(A) U

whenever A U and U is g*-open in (X, ).

Definition 2.9.1.15: A subset A of topological space X, is called

-closed set (Mahi et al. 1994), if cl (A) U whenever A U and U is

-open.

Definition 2.9.1.16: A subset A of topological space X, is called

gpr -closed set (Gnanambal 1997), if pcl(A) U whenever A U and U is

g-open in (X, ).

Definition 2.9.1.17: A subset A of topological space X is called wg

-closed set (Nagaveni 1999), if cl(int(A)) U whenever A U and U is open

in (X, ).

Definition:2.9.1.18: A subset A of topological space X, is called

swg-closed set (Nagaveni 1999), if cl(int(A)) U whenever A U and U is

semi-open in (X, ).

2.9.2 Separation Axioms

Definition 2.9.2.1: A space X is said to be semi- 1/ 2T -space

(Bhattacharya and Lahiri 1987), if every sg-closed set is semi-closed.

Definition 2.9.2.2: A space X is said to be pre- 1/ 2T space (Umhera

and Noiri 1996), if every gp-closed set is pre-closed.

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Definition 2.9.2.3: A space X is said to be semi-pre- 1/ 2T -space

(Dontchev 1995), if every g -closed set and gsp-closed set is -closed set

and semi-pre-closed set.

Definition 2.9.2.4: A space X is said to be 1/ 2T -space (Dunham

1977), if every g-closed set is closed or equivalently if every singleton is open

or closed.

Definition 2.9.2.5: A space X is said to be pre-regular- 1/ 2T -space

(Gnanambal 1997), if every gpr-closed set is pre-closed set. Note that a subset

A is called gpr-closed whenever pclA U whenever A U and U is regular

open.

Definition 2.9.2.6: A space X is said to be gT -space (Mahi et al.

1994), if every g -closed set is g-closed set.

Definition 2.9.2.7: A space X is said to be gT -space (Mahi et al.

1998), if every g-closed set is g -closed set.

Definition 2.9.2.8: A space X is said to be an dT -space (Mahi

et al. 1998), if every g-closed set is g-closed set.

Definition 2.9.2.9: A space X is said to be gsT -space (Mahi et al.

1993), if every gs-closed set is sg-closed.

2.9.3 Continuous Functions

Definition 2.9.3.1: Let X and Y be the topological spaces. A map

f : (X, (Y, ) is said to be generalized b-continuous maps (Ahmad Al

-omari and Mohd-Salmi Md.Noorani 2009), if the inverse image of every

open set in Y is gb-open in X.

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f : (X, (Y, ) is said to be generalized semi-continuous maps (Devi et al.

1993), if the inverse image of every open set in Y is gs-open in X.


f : (X, (Y, ) is said to be weakly generalized-continuous maps (Nagaveni

1999), if the inverse image of every open set in Y is wg-open in X.


f : (X, (Y, ) is said to be semi-weakly generalized-continuous maps

(Nagaveni 1999), if the inverse image of every open set in Y is swg-open in

X.


f : (X, (Y, ) is said to be generalized pre-continuous maps (Mashour

1982), if the inverse image of every open set in Y is gp-open in X.


f : (X, (Y, ) is said to be generalized -continuous maps (Mahi et al.

1994), if the inverse image of every open set in Y is g -open in X.


f : (X, (Y, ) is said to be generalized-continuous maps (Mahi et al.

1994), if the inverse image of every open set in Y is g-open in X.

2.9.4 Closed Maps and Open Maps


f : (X, (Y, ) is said to be g-closed (resp. g-open) (Levine 1970), if each

closed (resp. open set) F of X, f(F) is g-closed (resp. g-open) in Y.

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f : (X, (Y, ) is said to be semi-closed (resp. semi-open) (Levine 1963),

if each closed (resp. open set) F of X, f(F) is semi-closed (resp. semi-open) in

Y.


f : (X, (Y, ) is said to be pre-closed (resp. pre-open) (Sen 1993), if each

closed (resp. open set) F of X, f(F) is pre-closed (resp. pre-open) in Y.


f : (X, (Y, ) is said to be regular-closed (resp. regular-open) (Nour

1995), if each closed (resp. open set) F of X, f(F) is regular-closed (resp.

regular-open) in Y.


f : (X, (Y, ) is said to be -open (Mashour et al. 1983), if the open set

F of X, f(F) is -set in Y.

Definition 2.9.4.6: A space X is said to be -normal (Mashour

et al. 1983), if for every two disjoint closed subsets A and B of X, there exists

two disjoint -open sets U and V such that A U and B M.

Definition 2.9.4.7: A space X is said to be -regular (Mashour

et al. 1983), if for each closed set F of X and each x X-F, there exits disjoint

-open sets U and V such that x U and F V.


f : (X, (Y, ) is said to be generalized b irresolute (briefly gb-irresolute)

(Ganster and Steiner 2007) map, if the inverse image of every gb-closed set

in Y is gb-closed in X.

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f : (X, (Y, ) is said to be -irresolute (briefly -irresolute) (Mashour

et al. 1983) map, if the inverse image of every -closed set in Y is -

closed in X.


f : (X, (Y, ) is said to be pre - irresolute (briefly p -irresolute) (Sen 1993)

map, if the inverse image of every pre -closed set in Y is pre -closed in X.

2.9.5 Homeomorphism

Definition 2.9.5.1: Let X and Y be the topological spaces. A

bijection map f : (X, (Y, ) is called generalized-homeomorphism (briefly

g-homeomorphism) (Levine 1970), if f and f-1

are g-continuous.


bijection map f : (X, (Y, ) is called generalized semi-homeomorphism

(briefly gs -homeomorphism) (Devi et al. 1993), if f and f-1

are gs-

continuous.

Definition 2.9.5.3: Let X and Y are topological spaces. A bijection

map f : (X, (Y, ) is called weakly generalized-homeomorphism

(briefly wg-homeomorphism) (Nagaveni 1999), if f and f-1

are wg-

continuous.


bijection map f : (X, (Y, ) is called semi-weakly generalized-

homeomorphism (briefly swg -homeomorphism) (Nagaveni 1999), if f and f-1

are swg-continuous.


bijection map f : (X, (Y, ) is called generalized -homeomorphism

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(briefly g -homeomorphism) (Mahi et al. 1994), if f and f-1

are g -

continuous.


bijection map f: (X, (Y, ) is called generalized-homeomorphism

(briefly g -homeomorphism) (Mahi et al. 1994), if f and f-1

are g-

continuous.


bijection map f : (X, (Y, ) is called generalized b- homeomorphism

(briefly gb-homeomorphism) (Ahmad Al-omari and Mohd-Salmi Md.Noorani

2009), if f and f-1

are gb-continuous.

2.9.6 Bitopological Spaces

Definition 2.9.6.1: A subset A of a bitopological space X, is called

( i, j) -weakly generalized closed (briefly ( i, j),-wg-closed) (Nagaveni

1999), if j-cl( i-int(A)) G whenever A U and U i.

Definition 2.9.6.2: A subset A of a bitopological space (X, 1, 2)

is said to be ( i, j)-regular open (Banerjee 1987), if A = i –int( j-cl(A))

where i, j = 1, 2 and i j.


is said to be ( i, j)-regular closed (Banerjee 1987), if A = i –cl( j-int(A))

where i, j = 1, 2 and i j.


is said to be ( i, j)-semi open (Bose 1981), if A j –cl( i -int(A)) where

i, j = 1, 2 and i j.

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is said to be ( i, j)-pre open (Jelic 1990), if A i –int( j-cl(A)) where i, j

= 1, 2 and i j.


is said to be ( i, j)-pre-closed (Jelic 1990), if i –cl( j-int(A)) A where

i, j = 1, 2 and i j.

Definition 2.9.6.7: A subset A of a bitopological space X is called

( i, j) -generalized closed (briefly ( i, j)-g-closed ) (Fukutake 1989), if

j-cl(A) G whenever A U and U i.


( i, j) -generalized *-closed (briefly ( i, j)-g*-closed ) (Ravi and Lellis

Thivagar 2011), if j-cl(A) U whenever A U and U GO(X, ).


( i, j)–semi-weakly generalized-closed (briefly ( i, j)-swg-closed)

(Nagaveni 1999), if j-cl(int(A)) U whenever A U and i G is i is

semi-open.

Definition 2.9.6.10: A subset A of a bitopological space X, is

called ( i, j)-semi-generalized-closed (briefly ( i, j)-sg-closed) (Khedr

and Al-Saadi 2009), if j-scl((A)) U whenever A U and i U then i is

semi-open.

Definition 2.9.6.11: A subset A of a bitopological space X, is

called ( i, j) -generalized semi-closed (briefly ( i, j)-gs-closed) (Khedr

and Al-Saadi 2009), if j-scl((A)) U whenever A U and i. U then i

is open.

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( i, j)-generalized b-closed (briefly ( i, j)-gb-closed) (Ganster and Steiner

2007), if j-bcl((A)) U whenever A U and i U then i is open.


( i, j) - strongly generalized -closed (briefly ( i, j)-gs-closed) (Al-Saadi

and Zakari 2008), if j-scl((A)) U whenever A U and U ( i, j)-g

-open.

Definition 2.9.6.14: A bitopological space (X, i, j) is said to be

a ( i, j) -*T½-space (Sheik John and Sundaram 2004), if every ( i, j)-g

-closed set and ( i, j)-g*-closed .


a ( i, j) -semi-T½-space (Khedr and Al-Saadi 2009), if every( i, j)-sg

-closed set is j -semi-closed set.


a ( i, j) -Tb-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed set

is j -closed set.


a ( i, j) -Td-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed set

is ( i, j) -g-closed set.


a ( i, j) -Tp-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed

set is j-closed set in X.

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a ( i, j) -Ts-space (Al-Saadi and Zakari 2008), if every ( i, j)-g-closed set

is ( i, j) - gs -closed set in X.

Definition 2.9.6.20: The kernel of a set A denoted by A is the

intersection of all the super sets of A (Mahi 1986) is A {U:A U,U }

Definition 2.9.6.21: A bitopological space (X, 1, 2) is called pair

wise - Pre -T2 ( Noiri and Popa 2007), if for each pair of distinct points x and

y of X there exist a ( i, j)- pre-open set U, containing x and ( j, i)-pre-

open set V containing y such that U V= for i, j=1, 2

Definition 2.9.6.22: A function f : (X, 1, 2) (Y, 1, 2) is said

to be Almost-S- continuous (Khedr and Noiri 2007), if for each x X and each

i-open set V containing f(x) there exist an ( i, j)-open set U of X

containing x such that f(U)i-int ( j -cl(V)).

Definition 2.9.6.23: A bitopological space (X, 1, 2) is said to be

pair wise semi-Uryshon (Khedr and Noiri 2007), if for each distinct points x

and y of X there exist a i -semi – open set U and a j -semi open set V such

that x U, y V and ( j -scl(U)) ( i -scl(V))= for i j, i, j=1,2.

Definition 2.9.6.24:: A bitopological space (X, 1, 2) is said to be

pair wise connected (Khedr and Noiri 2007), if it cannot be expressed as the

union of two non-empty disjoint sets U and V such that, U is i-open and V is

j-open.

Definition 2.9.6.25: A bitopological space (X, 1, 2) is said to be

pair wise semi-connected (Khedr and Noiri 2007), if it cannot be expressed as

the union of two semi-separated sets called a semi-connected sets U and V

such that U is i-semi open and V is j-semi open.

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Definition 2.9.6.26: A point x of X is called an ( i, j)- -cluster

point of A (Khedr and Noiri 2007), if i -int( j-cl(U)) A= for every i

-open set U containing x.

Remark 2.9.6.27: The set of all ( i, j)- -cluster points of A is

called ( i, j)- -cluster of A (Khedr and Noiri 2007) and is denoted by

( i, j) - cl (A).

Definition 2.9.6.28: A point x of A is called an ( i, j)- -interior

point of A denoted by ( i, j)- int (A) (Khedr and Noiri 2007), if there

exists U i, such that x U i-int( j-cl(U)) A.

Definition 2.9.6.29: A point x of X is called an ( i, j)- -cluster

point of A (Khedr and Noiri 2007), if j-(cl(U)) A for every i-open

set U containing x.

Remark 2.9.6.30: The set of all ( i, j)- -cluster points of A is

called ( i, j)- -closer of A (Khedr and Noiri 2007) and is denoted by

( i, j) - cl (A).

Definition 2.9.6.31: A point x of A is said to be ( i, j)

- -interior point of A (Khedr and Noiri 2007) denoted by ( i, j)- int (A),

if there exist U i such that x U j -cl(U) A.

Definition 2.9.6.32: A subset A of a space X is called ( i, j)-semi

regular (Khedr and Noiri 2007), if A is both ( j, i)-semi-open and ( i, j)

-semi closed. The family of all the ( i, j)-semi-regular sets of a space X is

denoted by ( i, j)-SR(X) and those containing x are denoted by ( i, j)

-SR(X, x)

33

Definition 2.9.6.33: A subset A of a space X is said to be ( i, j)

-open (Khedr and Noiri 2007), if A i-int( j-cl( i-int(A)). The family of

all ( i, j)- -open sets of a space X is denoted by ( i, j)- (X) and those

containing x are denoted by ( i, j)- (X, x)

2.9.7 Fuzzy Topological Spaces

Fuzzy Sets 2.9.7.1: Let X be a non-empty set . A fuzzy set

in X is a function from X into the closed unit interval [0,1]. When if

(x) (x) for all x X. = , means and . If { i : i I} is a

collection of fuzzy sets in X, then i and i are given by( i) (x) =

sup{ i(x): i I} for all x X and ( i) = inf { i(x): i I} for all x X. The

complement c of a fuzzy set in X is given by

c(x) = 1- (x) for all

x X. The fuzzy sets 0 and 1 are given by 0(x) = 0 and 1(x) = 1 for all x X.

Definition 2.9.7.2 (Chang 1968) : A fuzzy topology on a set X is

collection of fuzzy sets such that,

a) 0,1

b) , then

c) If i , then i

Members of are called fuzzy open sets and the pair (X, ) is

called a fuzzy topological space. Complements of fuzzy open sets are called

fuzzy closed sets. The closure denoted by cl( ) and interior denoted by int( )

of a fuzzy set in a fuzzy topological space are given by cl( ) = { : is a

fuzzy closed set } and int( ) = { : is a fuzzy open set }

It is noted that cl( ) is the smallest fuzzy closed set containing

and int( ) is the largest fuzzy open set contained in . Also, cl( ) =

cl( ) cl( ), int(1- ) = 1-cl( ) and cl(1- ) = 1- int( ).

34

Definition 2.9.7.3: A fuzzy set in a fuzzy topological space X is

said to be fuzzy semi-open, if there exists a fuzzy open (Azad 1981) set V in

X such that V cl(V).

Definition 2.9.7.4: A fuzzy set in a fuzzy topological spaces X is

said to be fuzzy semi –closed, if and only if there exists a fuzzy closed (Azad

1981) set V in X such that int(V) V.

Remark 2.9.7.5: It is seen that a fuzzy set is fuzzy semi-open

(Azad 1981), if and only if c

is fuzzy semi-closed.

Definition 2.9.7.6: A fuzzy set of a fuzzy topological space X is

called fuzzy regular open set (Azad 1981) of X, if int(cl( )) = .

Remark 2.9.7.7: It is seen that a fuzzy set is fuzzy regular open

(Azad 1981), if and only if c is fuzzy regular closed set of X, if cl(int( )= .

Definition 2.9.7.8: Let X be a fuzzy topological space, a fuzzy set

in X is then called a fuzzy -open set (Azad 1981), if

int(cl(int( )).

Definition 2.9.7.9: Let X be a fuzzy topological space, then a fuzzy

set in X is called a fuzzy- -closed set (Bin shahana 1991), if cl(int(cl( ))

.


in X is then called a fuzzy pre-open set (Singal 1991), if int(cl(( )).


in X is then called a fuzzy pre-closed (Singal 1991), if cl(int( )) .

35

Definition 2.9.7.12: A fuzzy set of a space (X, ) is called fuzzy

generalized -closed (briefly fuzzy g -closed) (Mahi et al. 1994), if cl )

whenever and is fuzzy -open.


generalized closed set (briefly fuzzy g-closed) (Balasubramanian and

Sundaram 1991), if cl ( ) whenever and -is fuzzy-open.


- generalized closed set (briefly fuzzy g-closed) (Mahi et al. 1994), if

cl ) whenever and -is fuzzy-open.


generalized semi-closed set (briefly fuzzy gs-closed) (Devi et al. 1993), if

scl( ) whenever and -is fuzzy -open.


generalized b-closed set (briefly fuzzy gb-closed) (Ahmad Al-omari and

Mohd-Salmi Md.Noorani 2009), if bcl( ) whenever and is fuzzy

–open.


generalized pre-closed set (briefly fuzzy gp-closed) (Mahi et al. 1996), if

pcl ( ) whenever and is fuzzy -open.


semi-generalized closed set (briefly fuzzy sg-closed) (Bhattacharya et al.

1987), if scl( ) whenever and -is fuzzy -open.

36


generalized semi pre-closed set (briefly fuzzy gsp-closed) (Dontchev 1995),

if spcl ( ) whenever and is fuzzy -open.


semi-weakly generalized closed set (briefly fuzzy swg-closed) (Nagaveni

1999), if cl(int( ) whenever and is fuzzy semi-open.

Definition 2.9.7.21: Let X and Y be the two fuzzy topological

spaces. A function f : (X, (Y, ) is called fuzzy g-continuous (briefly fg-

continuous) (Levine 1970), if f-1

) is fg-closed set in X, for every fuzzy

closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy generalized pre-

continuous (briefly fgp-continuous) (Mahi et al. 1996), if f-1

) is fgp-closed

set in X, for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy semi-generalized-

continuous (briefly fsg-continuous) (Bhattacharya et al. 1987), if f-1

) is fsg-

closed set in X, for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy generalized semi-

continuous (briefly fgs -continuous) (Devi et al. 1993), if f-1

) is fgs-closed

set in X, for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy semi-weakly generalized-

37

continuous (briefly fswg-continuous) (Nagaveni 1999), if f-1

) is fswg-

closed set in X, for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy generalized b-continuous

(briefly fgb-continuous) (Ahmad Al-omari and Mohd-Salmi Md.Noorani

2009), if f-1

) is fgb-closed set in X, for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) fuzzy generalized pre-regular-continuous

(briefly fgpr-continuous) (Gnanambal 1997), if f-1

) is fgpr-closed set in X,

for every fuzzy closed set of Y.


spaces. A function f : (X, (Y, ) is called fuzzy generalized-continuous

(briefly f g-continuous) (Mahi et al. 1994), if f-1

) is f g-closed set in X,

for every fuzzy closed set of Y.


spaces. A bijection map f : (X, (Y, ) is called fuzzy weakly generalized-

homeomorphism (briefly fwg-homeomorphism) (Nagaveni 1999), if f and

f-1

are fwg-continuous maps.


spaces. A bijection map f : (X, (Y, ) is called fuzzy semi-weakly

generalized-homeomorphism (briefly fswg-homeomorphism) (Nagaveni

1999), if f and f-1

are fswg-continuous maps.


spaces. A bijection map f : (X, (Y, ) is called fuzzy generalized b-

38

homeomorphism (briefly fgb-homeomorphism) (Ahmad Al-omari and Mohd-

Salmi Md.Noorani 2009), if f and f-1

are fgb-continuous maps.


spaces. A bijection map f : (X, (Y, ) is called fuzzy generalized semi-

homeomorphism (briefly fgs-homeomorphism) (Devi et al. 1993), if f and

f-1

are fgb-continuous maps.


spaces. A bijection map f : (X, (Y, ) is called fuzzy generalized -

homeomorphism (briefly fg -homeomorphism) (Mahi et al. 1994), if f and

f-1

are fg -continuous maps.


spaces. A bijection map f : (X, (Y, ) is called fuzzy generalized-

homeomorphism (briefly f g-homeomorphism) (Mahi et al. 1994), if f and

f-1

are f g-continuous maps.

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