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13
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.
14
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.
15
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),
16
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.
(3) On generalized b-closed sets:
Ahmad Al-omari and Mohd.Salmi Md.Noorani (2009).
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).
17
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.
(3) On generalized b-closed sets:
Ahmad Al-omari and Mohd.Salmi Md.Noorani (2009).
The authors proved that if the bijective f : X Y is b-irresolute
and open, f is then gb-irresolute.
18
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).
19
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.
20
(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
21
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)).
22
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.
23
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.
24
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.
25
Definition 2.9.3.2: Let X and Y be the topological spaces. A map
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.
Definition 2.9.3.3: Let X and Y be the topological spaces. A map
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.
Definition 2.9.3.4: Let X and Y be the topological spaces. A map
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.
Definition 2.9.3.5: Let X and Y be the topological spaces. A map
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.
Definition 2.9.3.6: Let X and Y be the topological spaces. A map
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.
Definition 2.9.3.7: Let X and Y be the topological spaces. A map
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
Definition 2.9.4.1: Let X and Y be the topological spaces. A map
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.
26
Definition 2.9.4.2: Let X and Y be the topological spaces. A map
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.
Definition 2.9.4.3: Let X and Y be the topological spaces. A map
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.
Definition 2.9.4.4: Let X and Y be the topological spaces. A map
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.
Definition 2.9.4.5: Let X and Y be the topological spaces. A map
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.
Definition 2.9.4.8: Let X and Y be the topological spaces. A map
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.
27
Definition 2.9.4.9: Let X and Y be the topological spaces. A map
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.
Definition 2.9.4.10: Let X and Y be the topological spaces. A map
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.
Definition 2.9.5.2: Let X and Y be the topological spaces. A
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.
Definition 2.9.5.4: Let X and Y be the topological spaces. A
bijection map f : (X, (Y, ) is called semi-weakly generalized-
homeomorphism (briefly swg -homeomorphism) (Nagaveni 1999), if f and f-1
are swg-continuous.
Definition 2.9.5.5: Let X and Y be the topological spaces. A
bijection map f : (X, (Y, ) is called generalized -homeomorphism
28
(briefly g -homeomorphism) (Mahi et al. 1994), if f and f-1
are g -
continuous.
Definition 2.9.5.6: Let X and Y be the topological spaces. A
bijection map f: (X, (Y, ) is called generalized-homeomorphism
(briefly g -homeomorphism) (Mahi et al. 1994), if f and f-1
are g-
continuous.
Definition 2.9.5.7: Let X and Y be the topological spaces. A
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.
Definition 2.9.6.3: A subset A of a bitopological space (X, 1, 2)
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.
Definition 2.9.6.4: A subset A of a bitopological space (X, 1, 2)
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.
29
Definition 2.9.6.5: A subset A of a bitopological space (X, 1, 2)
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.
Definition 2.9.6.6: A subset A of a bitopological space (X, 1, 2)
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.
Definition 2.9.6.8: A subset A of a bitopological space X, is called
( 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, ).
Definition 2.9.6.9: A subset A of a bitopological space X, is called
( 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.
30
Definition 2.9.6.12: A subset A of a topological space X, is called
( 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.
Definition 2.9.6.13: A subset A of a topological space X, is called
( 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 .
Definition 2.9.6.15: A bitopological space (X, i, j) is said to be
a ( i, j) -semi-T½-space (Khedr and Al-Saadi 2009), if every( i, j)-sg
-closed set is j -semi-closed set.
Definition 2.9.6.16: A bitopological space (X, i, j) is said to be
a ( i, j) -Tb-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed set
is j -closed set.
Definition 2.9.6.17: A bitopological space (X, i, j) is said to be
a ( i, j) -Td-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed set
is ( i, j) -g-closed set.
Definition 2.9.6.18: A bitopological space (X, i, j) is said to be
a ( i, j) -Tp-space (Al-Saadi and Zakari 2008), if every ( i, j)-gs-closed
set is j-closed set in X.
31
Definition 2.9.6.19: A bitopological space (X, i, j) is said to be
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( ))
.
Definition 2.9.7.10: Let X be a fuzzy topological space, a fuzzy set
in X is then called a fuzzy pre-open set (Singal 1991), if int(cl(( )).
Definition 2.9.7.11: Let X be a fuzzy topological space, a fuzzy set
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.
Definition 2.9.7.13: A fuzzy set of a space (X, ) is called fuzzy
generalized closed set (briefly fuzzy g-closed) (Balasubramanian and
Sundaram 1991), if cl ( ) whenever and -is fuzzy-open.
Definition 2.9.7.14: A fuzzy set of a space (X, ) is called fuzzy
- generalized closed set (briefly fuzzy g-closed) (Mahi et al. 1994), if
cl ) whenever and -is fuzzy-open.
Definition 2.9.7.15: A fuzzy set of a space (X, ) is called fuzzy
generalized semi-closed set (briefly fuzzy gs-closed) (Devi et al. 1993), if
scl( ) whenever and -is fuzzy -open.
Definition 2.9.7.16: A fuzzy set of a space (X, ) is called fuzzy
generalized b-closed set (briefly fuzzy gb-closed) (Ahmad Al-omari and
Mohd-Salmi Md.Noorani 2009), if bcl( ) whenever and is fuzzy
–open.
Definition 2.9.7.17: A fuzzy set of a space (X, ) is called fuzzy
generalized pre-closed set (briefly fuzzy gp-closed) (Mahi et al. 1996), if
pcl ( ) whenever and is fuzzy -open.
Definition 2.9.7.18: A fuzzy set of a space (X, ) is called fuzzy
semi-generalized closed set (briefly fuzzy sg-closed) (Bhattacharya et al.
1987), if scl( ) whenever and -is fuzzy -open.
36
Definition 2.9.7.19: A fuzzy set of a space (X, ) is called fuzzy
generalized semi pre-closed set (briefly fuzzy gsp-closed) (Dontchev 1995),
if spcl ( ) whenever and is fuzzy -open.
Definition 2.9.7.20: A fuzzy set of a space (X, ) is called fuzzy
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.
Definition 2.9.7.22: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.23: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.24: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.25: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.26: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.27: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.28: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.29: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.30: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.31: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.32: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.33: Let X and Y be the two fuzzy topological
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.
Definition 2.9.7.34: Let X and Y be the two fuzzy topological
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.