CHAPTER 2 LITERATURE SURVEY

Transcription

1 17 CHAPTER 2 LITERATURE SURVEY 2.1 INTRODUCTION Topology is the study of connectedness and continuity. A topological space is less well defined than our everyday metric spaces, and includes no notion of strict distances. Instead, it uses the concept of open sets to indicate the nearness of certain points. It also discusses the idea of homeomorphism, and how two topological spaces are essentially the same under few conditions. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension and transformation. 2.2 CLOSED SETS AND OPEN SETS IN TOPOLOGICAL SPACES Regular open sets and strong regular open sets have been introduced and investigated by Stone (1937) and Tong (1982) respectively. Andrijevic (1996) introduced generalized open sets in topological spaces called b-open sets. Maki et al (1999) studied on generalizing semi-open sets and pre-open sets in topological spaces. Benchalli et al (2009) introduced a class of - closed sets. Mugarjee & Sinha (2008) introduced S-cluster sets and S- closedness in topological spaces. A class of sets called #g-semi closed sets was introduced by Veerakumar (2005). The following list gives a brief account of work done in topological spaces.

2 18 (1) Generalized closed set in topology: Levin (1970) The author introduced g-closed sets and studied some properties of g-closed sets in normal spaces, complete uniform spaces, locally compact Hausdorff spaces. Relationships between T 1 -space and T 1/2 space were also presented in this paper. (2) Generalized - closed sets in topological spaces. Maki et al (1993) They introduced two classes of generalized -closed sets and investigated some basic properties of those sets. The following implication relations are proved in this paper. A is g *closed in ) A is g **closed in. (3) Strongly generalized closed sets in topological spaces: Sundaram & Pushpalatha (2001) In this paper they introduced the concepts of strongly generalized closed sets and strongly generalized open sets, which are generalizations of closed sets and open sets in topological spaces. Further they introduced T S and T P spaces. Also introduced a closure operator C S and obtained topology S and studied some of their properties.

3 19 (4) On RW-closed sets in topological spaces. Benchalli & Wali (2007) The authors introduced a class of sets called regular w-closed (briefly rw-closed) sets in topological spaces. A subset A of a topological -closed if U contains closure of A whenever U contains A, U is regular semi the class of all w-closed sets and the class of all regular generalized closed sets. Also they investigated some of their properties. (5) -closed s -open sets in topological spaces Vadivel & Vairamanickam (2009) In this paper, they introduced -closed sets, -open sets and some of their basic properties are brought out. They proved that the class of -closed sets lies between the class of -closed sets and the class of regular generalized closed sets, every w- -closed set in X, but not conversely, -closed set in X is rg-closed set in X, but not conversely, -closed but not conversely and - closed set in X is rwg-closed set in X, but not conversely. 2.3 SEPARATION AXIOMS IN TOPOLOGICAL SPACES Maki et al (1996) introduced pre- T 1/2 -space in topological spaces. Devi et al (1998) introduced a spaces called T d -spaces and T b -spaces and investigated the relationship among these spaces T i where i = 1, 1/2. They proved the concepts of T d -spaces and T b -spaces are preserved under homeomorphism. Araki et al (2006) studied characterization of semi-t i -

4 20 spaces. Brief analyses of some of the articles published that are related to this topic are givrn below. (1) Remarks on semi generalized and generalized semi closed sets Maki et al (1996) The authors introduced a T gs -space and investigated some of its properties in this paper. They discussed some relationships of this space with few existing spaces. (2) Studies on generalization of homeomorphisms in topological spaces: Nagaveni N (1999) The author introduced T wg -spaces and T swg -spaces and investigated their relationships with few other spaces. (3) Properties of T 1/4 spaces: Araki et al (2006) The authors introduced T 1/4 spaces in their study of generalized continuity and -closed sets. Also, the product of T 1/4 spaces is studied. The authors proved the followings, T 1/4 space. Then every subspace of X is a T 1/4 1/4 space, (b) For every locally finite subset F X and every point y F there exists a subset A X such that F A, y A and A is open or closed.

5 CLOSURE OPERATOR IN TOPOLOGICAL SPACES Dunhum (1982) introduced the concept of the closure operator cl* of a set in a topological space (X, ) and a topology * and studied some of their properties using g-closed sets. He proved that cl* operator is a Kuratowski closure operator. Sundaram et al (1991) introduced a class of semi generalized closure operator in topological spaces. Here are some other closure operators introduced by different authors. (1) Generalized -sets and the associated closure operator: Haruo Maki (1986) The author introduced the concept of a generalized -sets in topological spaces and defined the associated closure operator c on the spaces and studied some of their properties. He also proved that c is a -open for every x in a topological (2) Weak form on g -closed sets, where { and Digital Planes Devi et al (2004) They introduced and studied weak form of generalized -closed sets in a topological space where. Further they studied wg c- homeomorphisms and their groups containing the homeomorphisms, where

6 CLOSED AND OPEN FUNCTIONS IN TOPOLOGICAL SPACES Generalized closed maps were introduced and studied by Malghan (1982), Noiri (1973, 1984), Mashour et al (1982, 1983), Crosssley et al (1972), Devi et al (1993, 1998), Noiri et al (1998), Arochiarani (1997) defined and studied semi-open maps, 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, g-closed maps and rg-closed maps respectively. Ptak (1958) studied on completeness and open mapping theorem. Nagaveni (1999) introduced the concepts of wg-closed and swgclosed maps in topological spaces and they discussed those map s relationship with some other closed maps and open maps. (Nasef 2005) introduced and studied contra -closed functions. Some studies on closed maps and open maps are given below: (1) Generaliz - -generalized closed maps: Devi et al (1998) The author introduced and investigated the concept of generalized - -generalized closed maps and -regular spaces as generalization of closed maps, generalized closed maps and regular spaces respectively. By introducing the concept of pre- -closed maps, it is proved -regularity are preserved under these maps. (2) RW-Closed maps and RW-Open maps in topological spaces Karpagadevi & Pushpalatha (2013)

7 23 In this paper they introduced rw-closed map from a topological space X in to a topological space Y as the image of every closed set is rwclosed and also they prove that the composition of two rw-closed maps need not be an rw-closed map and obtained some properties of rw-closed maps. 2.6 CONTINUOUS FUNCTIONS IN TOPOLOGICAL SPACES Several topologists introduced the weak and strong forms of continuous maps. Balachandran et al (1991), Dontchev (1996), Arockiarani (1997), Ganster & Reilly (1989), Levin (1961), Maheswari & Thakur (1985), Mashour et al (1982), Nasef (2001), Biswas (1969), Ganster & Reilly (1989), Noiri (1974), Mashour et al (1982), Tong (1986) and Devi et al (1993, 1998) introduced and discussed simple continuity, almost continuity, weak continuity, -continuity, -continuity, semi-weak continuity and weak almost continuity and -generalized continuity and generalized -continuity. Noiri (1974), Levin (1960), Arya & Gupta (1974) and Reilly & Vamanamurthy (1983) have introduced and discussed some strong forms of continuous maps. Also they introduced the concept of various maps like strong continuous, strongly - continuous, clopen continuous and super continuous respectively. (Ahmad Al-Omari et al 2009) investigated and studied generalized-b-continuous maps. Generalized-semi-pre-continuous (briefly gsp-continuous) maps are introduced by Dontchev (1995). Regular generalized continuous (briefly rg-continuous) maps are introduced by Park et al (1995). g Continuous maps are introduced by Maragathavalli & Sheik John (2005). Contracontinuous functions are presented by Dontchev (1996). Vinayagamoorthi & Nagaveni (2012) introduced and studied Continuous functions in topological spaces. From the above concepts, a short survey of the article published on continuous maps is given below.

8 24 (1) -continuous and -open mapping: Mashour et al (1981) Tthe authors established and studied -continuous and -open mapping in topological spaces. (2) On continuous maps in topological spaces Balachandran et al (1991) They introduced and studied the concept of a class of maps, namely g-continuous maps, which includes the class of continuous maps, and a class of gc-irresolute maps. Moreover they introduced the concept of GOcompactness and GO-connectedness of topological spaces and proved product theorems for GO-compact spaces and GO-connected spaces. (3) Semi generalized continuous maps and semi T 1/2 -spaces: Sundaram et al (1999) The authors introduced a class of semi generalized continuous maps further they investigate the semi-homeomorphic image of semi T 1/2 -spaces and studied the product Theorem for semi T 1/2 -spaces. (4) Contra rw-continuous Functions Vadivel et al (2011) In this paper they introduced the notion of contra rw-continuous functions and obtained fundamental properties of contra rw-continuous functions. Also, they discussed the relationship between contra rw-continuity and other related functions.

9 IRRESOLUTE MAPS IN TOPOLOGICAL SPACES Cammaroto (1989), Ganster & Reilly (1989), Maheswari & Thakur (1985), Dontchev (1996), Balachandran et al (1996), Devi et al (1998), Arokiarani (1997) introduced and studied almost irresolute functions, weak irresolute maps and -irresolute maps, -irresolute maps, gc-irresolute maps, - irresolute maps, -irresolute maps and gr-irresolute maps respectively. Faro (1987) studied strongly -irresolute mappings in topological spaces. Balachandran et al (1991) introduced a class of generalized irresolute maps in topological spaces and they studied their relationship with other irresolute maps. Nagaveni (1999) introduced a class of weakly generalized irresolute and semi weakly generalized irresolute maps in topological spaces. Ganster et al (2007) introduced generalized b-irresolute (briefly gb-irresolute) maps. Here, a short list of irresolute maps published in various journals is given. (1) - -Irresolute Functions Devamanoharan et al (2012) In this paper two types of irresolute - -irresolute functions are introduced and characterized. (2) On -irresolute maps, where {strongly, strongly semi, almost} Nono et al (2003) The authors introduced and studied the concept of strongly - irresolute functions and almost generalized functions in topological spaces.

10 HOMEOMORPHISMS IN TOPOLOGICAL SPACES Many topologists Arociarani (1997), Biswas (1969), Crossley (1972), Devi (1995), Maki (1991), Noiri (1984), Sivaraj (1986) and Sundaram (1991) generalized the notion of homeomorphisms. Semihomeomorphism which is weaker than a homeomorphism was introduced by Biswas(1969). Crossely & Hilderbrand (1972), Neubrun (1977), Neubrunnova (1973) and Piotrowski (1979) proved that semihomeomorphism of Biswas (1969) and semi homeomorphism of Crossley (1972) are independent. Semi-generalized homeomorphisms and generalized semi-homeomorphisms, somewhat homeomorphisms, -homeomorphism, -homeomorphism, -homeomorphism, g-homeomorphisms and gchomeomorphism, rg-homeomorphisms and (, )-homeomorphisms have been defined by Devi et al (1993), Gentry & Hoyle (1971), Tadros & Abd Allah (1990) Umehara & Maki (1989), Maki et al (1991), Arokiarani (1997) and Ogata (1991) respectively. Miguel Caldas (2001) studied about generalized homeomorphisms in topological spaces. Ahmad Al-Omari & Mohd-Salmi Md Noorani (2009) introduced generalized b-homeomorphism (gb-homeomorphism). Some studies on generalized homeomorphisms are listed below. (1) On generalized homeomorphism in topological spaces: Maki et al (1991) The authors defined a class of generalized homeomorphism and generalized c-homeomorphism which are generalization of homeomorphism and they investigated some properties of generalized homeomorphism. The they proved that ontains subgroup and gch

11 27 (2) Semi-generalized homeomorphisms and generalized semi homeomorphisms in topological spaces: Devi et al (1995) The authors introduced a class of semi generalized homeomorphisms and generalized semi homeomorphism in topological spaces and studied some of their properties. Moreover, some properties of these mapping from the quotient spaces to other spaces are also investigated. (3) Studies on generalization of homeomorphisms in topological spaces: Nagaveni (1999) The author introduced weakly generalized homeomorphisms and semi-weakly generalized homeomorphisms in topological spaces also discussed the relations with some existing homeomorphisms. (4) -Homeomorphisms in Topological Spaces Vadivel & Vairamanickam (2010) In this paper a class of -homeomorphisms and a class of - -homeomorphisms were introduced. It is also proved that set -homeomorphisms forms a group under the operation composition of maps.

12 COMPACTNESS AND CONNECTEDNESS IN TOPOLOGICAL SPACES SgC-compact was introduced by Abd EI-Monsef (1983). Kozae (1985) introduced -compact topological spaces. Balachandran et al (1991) introduced the concepts of generalized open compactness and generalized open connectedness in topological spaces and then proved the product theorems for generalized open compact spaces and generalized open connected spaces. -connected topological spaces are presented and studied by Nour (1995). Thereafter a large number of analyses have been done towards the concepts of compactness and connectedness in a topological space, few are listed below. (1) Note on semi-connectedness Das (1981) The author studied some of the properties of the semi-connectedness in topological spaces. (2) Semi generalized-homeomorphisms and generalized semihomeomorphisms in topological spaces. Devi et al (1995) The authors introduced a class of generalized semi open-compact, semi generalized open-compact sets and studied some of the properties for such spaces. The authors also proved, (i) A sg-closed subset of a SGOis also SGO-compact relative to X. (ii) A gs-closed subset of a GSO- GSO-compact relative to X.

13 29 (3) Some characterizations of generalized pre-irresolute and generalized pre-continuous maps between topological spaces: Arokiarani et al (1999) The authors introduced generalized pre-open-compact topological spaces and they studied some properties of such sets. sets in (4) On Semi*-Connected and Semi*-Compact Spaces Robert & Pious Missier (2012) In this paper they introduced the concepts of semi*-connected spaces, semi*-compact spaces and semi*-lindel of spaces and investigated some of their basic properties and their relationship with already existing concepts LOCALLY CLOSED SETS IN TOPOLOGICAL SPACES Kuratowski & Sierpioski (1921) introduced the notion of a locally closed set in a topological space using open sets and closed sets. Stone (1980, 1989) introduced and studied about absolutely FG spaces for a locally closed subset. Levine (1970) introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Using generalized closed sets and generalized open set, Balachandran, Sundaram & Maki (1996) defined and studied generalized locally closed sets, GLC-continuous function, Semi-generalized locally closed sets, and SLC-continuous function. Semigeneralized locally closed sets and SGLC continuous functions are introduced and studied by Jin Han Park and Jin Keun Park (2000). g*s-connectedness and g*s-locally closed sets in topological spaces are introduced and studied by Anitha & Pushpalatha (2012). Alli et al (2013)

14 30 studied g#p-locally closed sets and g#p-locally closed functions. Veera Kumar (2004) introduced and studied g#-locally closed sets and G#LCfunctions. Some of the research papers on locally closed sets are given below. (1) Locally closed sets and LC-continuous functions Ganster & Reilly (1989) In this paper they introduced and studied three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-lccontinuity. All three notions are defined by using the concept of a locally closed set. A subset S of a topological space X is locally closed if it is the intersection of an open and a closed set. Further some properties of these functions are studied and proved that a function between topological spaces is continuous if and only if it is sub-lc-continuous and nearly continuous in the sense of Ptak (1958). (2) Remarks on Locally closed sets Ganster et al (1992) This paper provides a useful characterization of LC(X, ), i.e. the family of locally closed subsets of (X, ), where denotes the -topology of a given topological space (X, ). In addition, various statements about the family of locally closed subsets of an arbitrary space and the relationships between these statements were examined. (3) Regular Generalized locally closed sets and RGL-Continuous functions Arokiarani et al (1997)

15 31 In this paper the authors introduced regular generalized locally closed sets and different notions of generalized continuous functions in a topological space and discussed some of their properties. (4) On Semi Generalized Locally Closed sets and SGLC continuous functions Jin Han Park & Keun Park (2000), In this paper sglc* sets and sglc** sets are introduced and different notions of continuous functions in a topological space are also introduced and their properties are investigated. (5) g*-locally closed sets and g*-lc Functions Veera Kumar (2003) In this paper, g*-locally closed sets and different notions of generalizations of continuous functions, namely, G*LC-continuity, G*LC*- continuity, G*LC* *-continuity, G*LC-irresoluteness, G*LC*-irresoluteness and G*LC* *-irresoluteness were introduced in a topological space BITOPOLOGICAL SPACE In 1963, J.C. Kelly initiated the study of bitopoloical spaces as a natural structure by studying quasi metrics and it s conjugate. This structure is a richer structure than of a topological space. Considerable effort had been expended in obtaining appropriate generalizations of standard topological properties to bitopological category by various authors. Khedr & Al-Saadi (2008) studied pairwise theta-semi-generalized closed sets. Selvanayaki (2012) studied g*s-closed sets in bitopological spaces. Few articles on bitopological spaces are listed below.

16 32 (1) On generalized closed sets in bitopological spaces, Fukutake (1985) Introduced the concept of generalized closed sets in bitopological spaces and studied some of their properties. (2) On weakly generalized closed sets, weakly generalized continuous maps and Twg-spaces in bitopological spaces, Fukutake et al (1999) The Authors Introduced the concept of weakly generalized closed sets, weakly generalized continuous maps and Twg-spaces in bitopological spaces and studied their properties. (3) g*-closed sets in bitopological spaces Sheik John & Sundaram (2004) In this paper g* closed sets in a bitopological spaces are introduced and its properties are studied, as an applications, two spaces (i,j)-t* ½ and (i,j)-*t 1/2 spaces are introduced and some of their propertied are studied. (4) - locally closed sets in Bitopological spaces, Benchalli Patil & Rayanagouder (2010) In this paper they -locally closed sets and -continuous maps in bitopological spaces and investigated some of their properties. (5) A study on generalizations of closed sets on continuous maps in topological spaces and in bitopological spaces Sheik John (2002)

17 33 The author defined a class of generalization called w-closed and he also introduced and studied about the concepts of w-locally closed sets and their corresponding continuous maps in topological space PRELIMINARIES work are given. In this section some basic definitions used for this proposed research Definition : A subset A of a topological space (X, ) is called regular open (Stone 1937) if A = int(cl(a)) and regular closed (Stone 1937) if A = cl(int(a)). Definition : A subset A of a topological space (X, ) is called pre-open (Mashhour et al 1982) if A int(cl(a)) and pre-closed (Mashhour et al 1982) if cl(int(a)) A. Definition : semiopen (Levine 1963) if A cl(int(a)) and semi-closed (Crossley & Hildebrand 1971) if int(cl(a)) A. The set of all semi-open sets in X is denoted by SO(X). scl(a) is the intersection of set all semi-closed sets in X containing A. Definition : -open (Njastad 1965) if A int(cl(int(a))) and -closed [Mashhour and Hasanein 1983] if cl(int(cl(a))) A. Definition : emipreopen (Andrijevic 1986) ( -open) if A cl(int(cl(a))) and semipreclosed (Abd El-Monsef et al 1983) ( -closed (1983)) if int(cl(int(a))) A.

18 34 Definition : A subset A of a topological space (X, -closed (Velicko1968) if A = cl (A), where cl (A) = {x X : cl(u) A, U and x U}. Definition : A subset A of a topological space (X, -closed (Velicko 1968) if A = cl (A), where cl (A) = {x X : int(cl(u)) A, U and x U}. Definition : A subset A of a topological space (X, generalized closed (briefly g-closed) (Levine 1970) if cl(a) U whenever A U and U is open in X. Definition : In a topological space (X, or the subset A of a topological space X, the generalized closure operator cl * (Dunhum 1982) is defined as the intersection of all g-closed sets containing A. Definition : In a topological space (X, or a subset A of a topological space X,the topology (Dunhum 1982) is defined by = {G:cl*(G c )=G c, G X and G is g-closed. Definition : (Dontchev & Noiri 2000) Let X be a topological space. The finite union of regular open sets in X is said to be -open. The complement of a -open set is said to be -closed. Definition : (Cameron 1978) A subset A of a topological space (X, ) is called regular semi open if there is a regular open set U such that U A cl(u). The family of all regular semi open sets of X is denoted by RSO(X). Remark : (Garg & Sivaraj 1984) If A is regular semi open in (X, ), then X A is also regular semi open.

19 35 Remark : (Garg & Sivaraj 1984) In a topological space (X, ), the regular closed sets, regular open sets and clopen sets are regular semi open. Definition : semigeneralized closed (briefly sg-closed) (Bhattacharya et al 1987) if scl(a) U whenever A U and U is a semi open set in X. Definition : generalized semi closed (briefly gs-closed) (Arya & Nour 1990) if scl(a) U, whenever A U and U is open in X. Definition : generalized closed (briefly rg-closed) (Palaniappan et al 1993) if cl(a) U, whenever A U and U is regular open in X. Definition : A subse - generalized closed (briefly g-closed) (Maki et al 1993) -cl(a) U, whenever A U and U is open in X. Definition : generalized -closed (briefly g -closed) (Maki et al 1994) if -cl(a) U, whenever A U and U is -open in X. Definition : generalized semi-pre-closed (briefly gsp-closed) (Dontchev1995) if spcl(a) U, whenever A U and U is open in X. Definition : A subset of a topolog generalized pre-closed (briefly gp-closed) (Maki et al 1996) if pcl(a) U, whenever A U and U is open in X.

20 36 Definition : -generalized closed (briefly -g-closed) (Dontchev & Ganster 1996) if cl (A) U, whenever A U and U is open in X. Definition : A subset A of a topological space(x, ) is called pre-generalized closed (briefly pg closed) set [Maki et al 1996] if pcl (A) U, whenever A U and U is pre- open in X. Definition : A topological space (X, ) is said to be pre-t ½ space (Umhera & Noiri 1996) if and only if every gp-closed set is pre-closed. Definition : generalized pre regular closed (briefly gpr-closed) (Gnanambal 1997) if pcl(a) U whenever A U and U is regular open in X. Definition : -generalized closed (briefly -g-closed) (Dontchev & Maki 1999) if cl (A) U, whenever A U and U is open in X. Definition : eakly generalized closed (briefly wg-closed) (Naganevi 1999) if cl(int(a)) U, whenever A U and U is open in X. Definition : weakly generalized closed (briefly swg-closed) (Nagaveni 1999) if cl(int(a)) U, whenever A U and U is semi open in X. Definition : weakly generalized closed (briefly rwg-closed) (Nagaveni 1999) if cl(int(a)) U whenever A U and U is regular open in X.

21 37 Definition : trongly generalized closed (Sundaram & Pushpalatha 2000) (briefly g*-closed) if cl(a) U, whenever A U and U is g-open in X. Definition : - generalized closed (briefly g-closed) (Dontchev & Maki 2000) if cl(a) U, whenever A U and U is -open in X. Definition : eakly closed (briefly w-closed) (Sundaram & Sheik John 2000) if cl(a) U, whenever A U and U is semi open in X. Definition : mildly generalized closed (briefly mildly g-closed) (Park 2004) if cl(int(a)) U, whenever A U and U is g-open in X. Definition : called regular w-closed (Benchalii & Wali 2007) (briefly rw-closed) if cl(a) U, whenever A U and U is regular semi open in (X, ).The set of all rw-closed sets in (X, is denoted by RWC(X). Remark : (Benchalii & Wali 2007) Let X be a topological space and A be a subset of X and let rsker(a) be the intersection of all the RSO sets containing A. If A is regular semi open in X, then rsker(a) = A, but not conversely. Remark : (Benchalii & Wali 2007) RSO(X) SO(X). Remark : (Benchalii & Wali 2007) For any subset A of (X, ), A rsker(a).

22 38 Remark : (Jankovic & Reilly 1985) Let x be a point of (X, ). Then {x} is either nowhere dense or preopen. Remark : (Benchalii & Wali 2007) Let A Y X, where X is a topological space and Y is an open subspace of X. If A RSO(X), then A RSO(Y). Remark : (Jankovic & Reilly 1985) The following decomposition of a given topological space (X, ), namely X = X 1 X 2 where X 1 = {x X : {x} is nowhere dense} and X 2 = {x X : {x} is preopen} is true. Remark : (Benchalii & Wali 2007) For any subset A of (X, ), X 2 cl(a) rsker(a), X 2 cl(a) rsker(a). Definition : (Balachandran et al 1991) said to be GO-connected if X cannot be written as a disjoint union of two non-empty g-open sets. A subset of X is GO-connected if it is GO-connected as a subspace. Definition : (Balachandran et al 1991) A collection {A i : i I} of g- open sets in a topological space X is called g-open cover(go-open cover) of a subset B if B {A i, i I }, where I is set all integers. Definition : (Balachandran et al 1991) A topological spaces X is g- compact if every GO-cover of X has a finite sub cover. Definition : Let X and Y be any two topological spaces. A map f : is called semi-closed [Levin 1963] map if for each closed set F of X, f(f)is semi-closed set in Y.

23 39 Definition : s called semi-continuous [Levin 1963] if f -1 (F) is semi-closed set in X for every closed set F in Y. Definition : Let X and Y be any two topological spaces. A bijection map f:(x, alled generalized-homeomorphism (briefly g- homeomorphism) [Levine 1970] if f and f -1 are g-continuous. Definition : Let X and Y be any two topological spaces. A map is called perfectly-continuous (Noiri 1984) if f -1 (F) is both open set and closed set in X for every open set F in Y. Definition : Let X and Y be any two topological spaces. A map closed (briefly g-closed) map (Sundaram 1991) if for each closed set F of X, f(f) is g-closed set in Y. Definition : Let X and Y be any two topological spaces. A map -open) map (Sundaram 1991) if for each open set F of X, f(f) is generalized open set in Y. Definition : Let X and Y be any two topological spaces. A map called generalized continuous (Balachandran et al 1991) if f -1 (F) is g-closed set in X for every closed set F in Y. Definition : Let X and Y be any two topological spaces. A map f: (X, -irresolute (briefly g-irresolute) map (Balachandran et al 1991) if f -1 (V) is a g-closed set in X for every g-closed set V in Y. Definition : Let X and Y be any two topological spaces. A map -irresolute (briefly gs-

24 40 irresolute) (Devi et al 1995) if f -1 (V) is a gs-closed in X for every gs-closed set V in Y. Definition : Let X and Y be any topological space. A map f is called weakly generalized continuous (briefly wgcontinuous) map (Nagaveni 1999) if f -1 (F) is wg-closed set in X for every closed set F in Y. Definition : from a topological space X into a topological Y is LC-continuous (Ganster & Reilly 1989) if f -1 (V) LC(X) for each open set V in Y. Definition : into a topological Y is LC-irresolute (Ganster & Reilly 1989) if f -1 (V) closed in X for each open set V in Y. Definition : A subset S of a topological space (X, closed (Ganster & Reilly1989) (briefly LC-closed) if S = A B where A is open and B is closed in X, the set of all LC-closed sets in X is denoted by LC(X). Definition : A subset S of a topological space (X, is called generalized locally closed set (Balachandran et al 1996) (briefly GLC) if S = A B, where A is g-open and B is g-closed in X. The set of all GLC-sets in X is denoted by GLC(X). Definition : A subset S of a topological space (X, is called GLC* (Balachandran et al 1996) if S = A B, where A is g-open and B is closed in X. The set of all GLC*-sets in X is denoted by GLC*(X).

25 41 Definition : A subset S of a topological space X is called GLC** (Balachandran et al 1996) if S = A B, where A is open and B is g-closed in X. The set of all GLC**-sets is denoted by GLC**(X). Definition : from a topological space X into a topological Y is GLC-continuous (Balachandran et al 1996) if f -1 (V) GLC(X) for each open set V in Y. Definition : from a topological space X into a topological Y is GLC-irresolute (Balachandran et al 1996) if f -1 (V) GLC(X) for each V in GLC(X). Definition : from a topological space X into a topological Y isglc*-continuous [1996] if f -1 (V) GLC*(X) for each open set V in Y. Definition : into a topological Y isglc**-continuous (Balachandran et al 1996) if f -1 (V) GLC**(X) for each open set V in Y. Definition : into a topological Y is GLC*-irresolute (Balachandran et al 1996) if f -1 (V) GLC*(X) for each V in GLC*(Y). Definition : into a topological Y isglc**-irresolute (Balachandran et al 1996) if f -1 (V) GLC**(X) for each V in GLC**(Y). Definition : A subset A of a bitopological space (X, i j) is called (i,j)-g-closed (Fukutake 1985) if j cl(a) U, whenever A U and U i.

26 42 Definition i j) is called (i,j)-rg-closed (Arockiarani 1997) j cl(a) U, whenever A U and U is i-regular open set. Definition i j) is called (i,j)-g*-closed (Sheik John & Sundaram 2004) j cl(a) U, whenever A U and U GO(X, i ). Definition i j) is called (i,j)-wg-closed (Fukutake et al 1999) j i-int(a)) U, whenever A U and U i. Definition i j) is called (i,j)-w-closed (Fukutake et al 2002) j cl(a) U, whenever A U and U i.-semi open set. Definition i j) is called (i,j)-gpr-closed (Fukutake et al 2002) j pcl(a) U, whenever A U and i-regular open set. Definition : (Fukutake 1985) A bitopological space (X, i be an (i,j)-t 1/2 -space if every (i,j)-g-closed set is j -closed. j )is said to Definition : (Fukutake et al 2002) A bitopological space (X,, i j )is said to be an (i,j)-t* 1/2 -space if every (i,j)-g*-closed set is j -closed. Definition : (Maki et al 1991) A function f: A function f:(x, i j ) (Y, i j) is called D-( i j )- k-continuous if the inverse image of every k- closed set is an ( i j )-g-closed set. Set of all ( i j )-g-closed sets is denoted by D( i j ) where k=1 or 2.

27 43 Definition : (Maki et al 1991) A function f:(x, i j ) (Y, i j) is called bi-continuous, if f is i i - j j continuous. Definition : (Maki et al 1991) A function f:(x, i j) (Y, i j) is called is strongly bi-continuous (s-bi-continuous), if f is i j -continuous and j i continuous. Definition : (Maki et al 1991) A function f:(x, i j) (Y, i j) is called generalized bi-continuous(g-bi-continuous), if f is D- i j)- j- continuous and D- j i)- i continuous. Definition : (Maki et al 1991) A function i j) i j) is called generalized strongly bi-continuous(g-s-bi-continuous), if f is g-bicontinuous, is D- i j)- i-continuous and D- j i)- j continuous. Definition : (Fukutake et al 2004) i j) i j) is called D*- i j)- k - k-closed set i j)-g*-closed set, where k=1 or 2. Definition : (Benchali & Wali 2007) i j) i, j i j) - i), f -1 (U) is rw- j) Definition : (Maki et al 1991) Let X =A B and let f:a Y and h:b Y be two functions. Any two such functions f and h are compatible if f /(A B) =h /(A B). Then, the following function f h:x Y is defined as follows (f h)(x)=f(x) for x A and (f h)(x)=h(x) for x B. The function (f h) is called combination of f and h. Definition : (Fukutake et al 2004) i j) i j) is called D*- i j)- k - k-closed set i j)-g*-closed set, where k=1 or 2.

28 44 Definition : (Benchali & Wali 2007) A function f:(x, i j ) (Y, i, j i j) - i), f -1 (U) is rw-open in (X, j ) Remark : (Fukutake 1985) Let (X, i j) be a bitopological space is i j)-t 1/2 if and only if {x} i j i closed for each x Definition : (Fukutake 1985) Let (X, i j) be a bitopological space. Then * i j)={e i j)-cl*(e c )=E c } is a topology on X. The detail studies of the above topics are established in the thesis.