On wgrα-continuous Functions in Topological Spaces

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1 Vol.3, Issue.2, March-Aprl pp ISSN: On wgrα-contnuous Functons n Topologcal Spaces A.Jayalakshm, 1 C.Janak 2 1 Department of Mathematcs, Sree Narayana Guru College, Combatore, TN, Inda 2 Department of Mathematcs, L.R.G.Govt.Arts.College for Women, Trupur, TN, Inda Abstract: In ths paper, we ntroduce new type of contnuous functons called strongly wgrα-contnuous and perfectly wgrαcontnuous and study some of ts propertes. Also we ntroduce the concept of wgrα-compact spaces and wgrα-connected spaces and some ther propertes are analyzed. Subject Classfcaton: 54C05, 54C10. Keywords: perfectly wgrα-contnuous, strongly wgrα-contnuous, wgrα-compact spaces and wgrα-connected spaces. I. Introducton Balachandran et al n [9, 10] ntroduced the concept of generalzed contnuous maps of a topologcal space. A property of gpr contnuous functons was dscussed by Y.Gnanambal and Balachandran K [5]. Strong forms of contnuty and generalzaton of perfect functons were ntroduced and dscussed by T.Nor [11, 12]. Regular α-open set s ntroduced by A.Vadvel and K. Varamanckam [14]. Rg-compact spaces and rg-connected spaces, τ * -generalzed compact spaces and τ * -generalzed connected spaces, gb-compactness and gb-connectedness ntroduced by A.M.Al.Shban [1], S.Eswaran and A.Pushpalatha [4], S.S.Benchall and Pryanka M.Bansal [2] respectvely. In ths paper we establsh the relatonshp between perfectly wgrα-contnuous and strongly wgrα-contnuous. Also we ntroduce the concept of wgrα-compact spaces and wgrα-connected spaces and study ther propertes usng wgrα-contnuous functons. II. Prelmnary Defntons Defnton: 2.1 A subset A of a topologcal space (X, τ) s called α-closed [10] f a nt (cl (nt (A)). Defnton:2.2 A subset A of a topologcal space (X, τ) s called gα-closed [9] f αcl (A) U,when ever A U and U s α-open n X. Defnton: 2.3 A subset A of a topologcal space (X, τ) s called rwg-closed [14] f cl (nt (A)) U, whenever A U and U s regular-open n X. Defnton: 2.4 A map f: X Y s sad to be contnuous [3] f f -1 (V) s closed n X for every closed set V n Y. Defnton: 2.5 A map f: X Y s sad to be wgrα- contnuous [6] f f -1 (V) s wgrα-closed n X for every closed set V n Y. Defnton: 2.6 A map f: X Y s sad to be perfectly-contnuous [12] f f -1 (V) s clopen n X for every open set V n Y. Defnton: 2.7 A map f: X Y s sad to be strongly-contnuous [8] f f -1 (V) s clopen n X for every subset V n Y. Defnton: 2.8 A functon f: X Y s called wgrα- rresolute [6] f every f -1 (V) s wgrα-closed n X for every wgrα-closed set V of Y. Defnton: 2.9 A functon f: X Y s sad to be wgrα-open [7] f f(v) s wgrα-open n Y for every open set V of X. Defnton: 2.10 A functon f: X Y s sad to be pre wgrα-open [7] f f(v) s wgrα-open n Y for every wgrα-open set V of X. Defnton: 2.11 A space (X,τ) s called wgrα-t 1\2 space[7] f every wgrα-closed set s α-closed. Defnton: 2.12 A space (X,τ) s called T wgrα -space[7] f every wgrα-closed set s closed. The complement of the above mentoned closed sets are ther respectve open sets. III. Strongly Wgrα-Contnuous and Perfectly Wgrα-Contnuous Functons Defnton: 3.1 A functon f :(X,τ) (Y,σ) s called strongly wgrα-contnuous f f -1 (V) s open n (X,τ) for every wgrα-open set V of (Y,σ). Defnton: 3.2 A functon f :(X,τ) (Y,σ) s called perfectly wgrα-contnuous f f -1 (V) s clopen n (X,τ) for every wgrα-open set V of (Y,σ). Defnton: Page

2 Vol.3, Issue.2, March-Aprl pp ISSN: A functon f :(X,τ) (Y,σ) s called strongly wgrα- rresolute f f -1 (V) s open n (X,τ) for every wgrα-open set V of (Y,σ). Defnton: 3.4 A functon f :(X,τ) (Y,σ) s called strongly rwg-contnuous f f -1 (V) s open n (X,τ) for every rwg-open set V of (Y,σ). Defnton: 3.5 A functon f :(X,τ) (Y,σ) s called perfectly rwg-contnuous f f -1 (V) s clopen n (X,τ) for every rwg-open set V of (Y,σ). Theorem: 3.6 If a functon f :(X,τ) (Y,σ) s perfectly wgrα-contnuous, then f s perfectly contnuous. Let F be any open set of (Y,σ).Snce every open set s wgrα-open.we get that F s wgrα-open n (Y,σ).By assumpton, we get that f -1 (F) s clopen n (X,τ). Hence f s perfectly contnuous. Theorem: 3.7 If f: (X,τ) (Y,σ) s strongly wgrα-contnuous, then t s contnuous. Let U be any open set n (Y,σ).Snce every open set s wgrα-open, U s wgrα-open n (Y,σ).Then f -1 (U) s open n (X,τ). Hence f s contnuous. Remark: 3.8 Converse of the above theorem need not be true as seen n the followng example. Example: 3.9 Let X={a,b,c,d},τ={ϕ,X,{a},{c,d},{a,c,d}} and σ={ϕ,y,{a},{b,c},{a,b,c}}. Defne f:x Y by f(a)=a, f(b)=d, f(c)=c, f(d)=b. Here f s contnuous, but t s not strongly wgrα-contnuous. Theorem: 3.10 Let (X,τ) be any topologcal space and (Y,σ) be a T wgrα -space and f: (X,τ) (Y,σ) be a map. Then the followng are equvalent: () f s strongly wgrα-contnuous. () f s contnuous. () () Let U be any open set n (Y,σ).Snce every open set s wgrα-open,u s wgrα-open n (Y,σ).Then f -1 (U) s open n (X,τ). Hence f s contnuous. ()() Let U be any wgrα-open set n (Y,σ).Snce (Y,σ) s a T wgrα -space,u s open n (Y,σ).Snce f s contnuous. Then f - 1 (U) s open n (X,τ). Hence f s strongly wgrα-contnuous. Theorem: 3.11 If f: (X,τ) (Y,σ) s strongly rwg -contnuous, then t s strongly wgrα- contnuous. Let U be any wgrα-open set n (Y,σ).By hypothess,f -1 (U) s open and closed n (X,τ). Hence f s strongly wgrα-contnuous. Remark: 3.12 Converse of the above theorem need not be true as seen n the followng example. Example: 3.13 Let X={a,b,c},τ=σ={ϕ,X,{a},{b},{a,b}}.Defne map f:x Y s an dentty map. Here f s strongly wgrα-contnuous, but t s not strongly rwg-contnuous. Theorem: 3.14 Let f: (X,τ) (Y,σ) be a map.both (X,τ) and (Y,σ) are T wgrα -space. Then the followng are equvalent: ()f s wgrα-rresolute. ()f s strongly wgrα-contnuous. () f s contnuous. (v) f s wgrα-contnuous. Straght forward. Theorem: 3.15 If f :(X,τ) (Y,σ) s strongly wgrα-contnuous and A s open subset of X, then the restrcton f A:A Y s strongly wgrα-contnuous. Let V be any wgrα-closed set of Y. Snce f s strongly wgrα-contnuous,then f -1 (V) s open n (X,τ).Snce A s open n X,(f A) -1 (V)=A f -1 (V) s open n A. Hence f A s strongly wgrα-contnuous. Theorem: 3.16 If a functon f :(X,τ) (Y,σ) s perfectly wgrα-contnuous, then f s strongly wgrα-contnuous. Let F be any wgrα-open set of (Y,σ).By assumpton, we get that f -1 (F) s clopen n(x,τ),whch mples that f -1 (F) s closed and open n (X,τ).Hence f s strongly wgrα-contnuous. Remark: 3.17 Converse of the above theorem need not be true as seen n the followng example. Example: Page

3 Vol.3, Issue.2, March-Aprl pp ISSN: Let X={a,b,c},τ={ϕ,X,{a},{c},{a,c}}= σ. Defne f:x Y by f(a)=a, f(b)=b, f(c)=c. Here f s strongly wgrα-contnuous, but t s not perfectly wgrα-contnuous. Theorem: 3.19 If f: (X,τ) (Y,σ) s perfectly rwg-contnuous, then t s perfectly wgrα-contnuous. As f s strongly contnuous,f -1 (U) s both open and closed n (X,τ) for every wgrα-open set U n (Y,σ). Hence f s perfectly wgrα-contnuous. Remark: 3.20 The above dscussons are summarzed n the followng dagram. perfectly rwg-contnuous perfectly wgrα-contnuous Perfectly contnuous strongly wgrα-contnuous Strongly rwg-contnuous contnuous Theorem: 3.21 Let (X,τ) be a dscrete topologcal space and (Y,σ) be any topologcal space.let f: (X,τ) (Y,σ) be a map. Then the followng statements are equvalent: () f s strongly wgrα-contnuous. () f s perfectly wgrα-contnuous. ()() Let U be any wgrα-open set n (Y,σ).By hypothess f -1 (U) s open n (X,τ).Snce(X,τ) s a dscrete space,f -1 (U) s also closed n (X,τ). f -1 (U) s both open and closed n (X,τ). Hence f s perfectly wgrα- contnuous. ()() Let U be any wgrα-open set n (Y,σ).Then f -1 (U) s both open and closed n (X,τ). Hence f s strongly wgrαcontnuous. Theorem:3.22 If f:(x,τ) (Y,σ) and g:(y,σ) (Z,μ) are perfectly wgrα-contnuous,then ther composton g f:(x,τ) (Z,μ) s also perfectly wgrα-contnuous. Let U be a wgrα-open set n (Z,μ).Snce g s perfectly wgrα-contnuous, we get that g -1 (U) s open and closed n (Y,σ).As any open set s wgrα-open n (X,τ) and f s also strongly wgrα-contnuous, f -1 (g -1 (U))= (g f) -1 (U) s both open and closed n (X,τ). Hence g f s perfectly wgrα-contnuous. Theorem:3.23 If f:(x,τ) (Y,σ) and g:(y,σ) (Z,μ) be any two maps. Then ther composton () wgrα-rresolute f g s strongly wgrα-contnuous and f s wgrα-contnuous. () Strongly wgrα-contnuous f g s perfectly wgrα-contnuous and f s contnuous. () Perfectly wgrα-contnuous f g s strongly wgrα-contnuous and f s perfectly wgrα-contnuous. ()Let U be a wgrα-open set n (Z,μ). Then g -1 (U) s open n (Y,σ).Snce f s wgrα-contnuous, g f: (X,τ) (Z,μ) s f -1 (g -1 (U))=(g f) -1 (U) s wgrα-open n (X,τ). Hence g f s wgrα-rresolute. ()Let U be any wgrα-open set n (Z,μ).Then g -1 (U) s both open and closed n (Y,σ) and therefore f -1 (g -1 (U))= (g f) -1 (U) s both open and closed n (X,τ). Hence g f s strongly wgrα-contnuous. () Let U be any wgrα-open set n (Z,μ).Then g -1 (U) s open and closed n (Y,σ).By hypothess, f -1 (g -1 (U)) s both open and closed n (X,τ).Hence g f s perfectly wgrα-contnuous. Theorem: 3.24 If f:(x,τ) (Y,σ) and g:(y,σ) (Z,μ) are strongly wgrα-contnuous,then ther composton g f:(x,τ) (Z,μ) s also strongly wgrα-contnuous. Let U be a wgrα-open set n (Z,μ).Snce g s strongly wgrα-contnuous, we get that g -1 (U) s open n (Y,σ).It s wgrα-open n (Y,σ) As f s also strongly wgrα-contnuous,f -1 (g -1 (U))= (g f) -1 (U) s open n (X,τ). Hence g f s contnuous. Theorem:3.25 If f:(x,τ) (Y,σ) and g:(y,σ) (Z,μ) be any two maps. Then ther composton g f :(X,τ) (Z,μ) s () strongly wgrαcontnuous f g s strongly wgrα-contnuous and f s contnuous. () wgrα-rresolute f g s strongly wgrα-contnuous and f s wgrα-contnuous. () Contnuous f g s wgrα-contnuous and f s strongly wgrα-contnuous. () Let U be a wgrα-open set n (Z,μ).Snce g s strongly wgrα-contnuous, g -1 (U) s open n (Y,σ).Snce f s contnuous, f -1 (g - 1 (U))= (g f) -1 (U) s open n (X,τ). Hence g f s strongly wgrα-contnuous. 859 Page

4 Vol.3, Issue.2, March-Aprl pp ISSN: () Let U be a wgrα-open set n (Z,μ).Snce g s strongly wgrα-contnuous, g -1 (U) s open n (Y,σ).As f s wgrα-contnuous, f - 1 (g -1 (U))= (g f) -1 (U) s wgrα-open n (X,τ). Hence g f s wgrα-rresolute. ()Let U be any open set n (Z,μ).Snce g s wgrα-contnuous, g -1 (U) s wgrα-open n (Y,σ).As f s strongly wgrαcontnuous, f -1 (g -1 (U))= (g f) -1 (U) s open n (X,τ). Hence g f s contnuous. Theorem: 3.26 Let f: (X,τ) (Y,σ) and g: (Y,τ) (Z,η) be two mappngs and let g f :(X,τ) (Z,η) be wgrα-closed.if g s strongly wgrαrresolute and bjectve, then f s closed. Let A be closed n (X,τ),then (g f)(a) s wgrα-closed n (Z,η).Snce g s strongly wgrα-rresolute,g -1 (g f)(a)=f(a) s closed n (Y,σ).Hence f(a) s closed. Theorem: 3.27 If f :(X,τ) (Y,σ) s perfectly wgrα-contnuous and A s any subset of X, then the restrcton f A:A Y s also perfectly wgrα-contnuous. Let V be any wgrα-closed set n(y,σ). Snce f s perfectly wgrα-contnuous, f -1 (V) s both open and closed n (X,τ).(f A) - 1 (V)=A f -1 (V) s both open and closed n A. Hence f A s perfectly wgrα-contnuous. IV. Wgrα-Compact Spaces Defnton: 4.1 A collecton {A α :αє } of wgrα-open sets n a topologcal space X s called wgrα-open cover of a subset B of X f B {A α :αє } holds. Defnton: 4.2 A topologcal space (X,τ) s wgrα-compact f every wgrα-open cover of X has a fnte subcover. Defnton: 4.3 A subset B of X s called wgrα-compact relatve of X f for every collecton {A α :αє } of wgrα-open subsets of X such that B {A α :αє },there exsts a fnte subset of such that B {A α :αє }. Defnton: 4.4 A subset B of X s sad to be wgrα-compact f B s wgrα-compact subspace of X. Theorem: 4.5 Every wgrα-closed subset of a wgrα-compact space s wgrα-compact space relatve to X. Let A be wgrα-closed subset of X, then A c s wgrα-open. Let O={G α :αє } be a cover of A by wgrα-open subsets of X.Then W=O A C s an wgrα-open cover of X. That s X=( {G α :αє }) A C.By hypothess, X s wgrα-compact. Hence W has a fnte subcover of X say (G 1 G 2 G 3 G n) A C. But A and A C are dsjont, hence A G 1 G 2 G n. So O contans a fnte subcover for A, therefore A s wgrα-compact relatve to X. Theorem: 4.6 Let f: X Y be a map: () If X s wgrα-compact and f s wgrα-contnuous bjectve, then Y s compact. ()If f s wgrα-rresolute and B s wgrα-compact relatve to X, then f(b) s wgrα-compact relatve to Y. ()Let f: X Y be an wgrα-contnuous bjectve map and X be an wgrα-compact space. Let {A α :αє } be an open cover for Y.Then {f -1 (A α ) :αє } s an wgrα-open cover of X. Snce X s wgrα-compact, t has fnte subcover say {f -1 (A 1 ),f - 1 (A 2 ),, f -1 (A n )},but f s surjectve,so {A 1,A 2,,A n } s a fnte subcover of Y. Therefore Y s compact. () Let B X be wgrα-compact relatve to X,{A α :αє } be any collecton of wgrα-open subsets of Y such that f(b) {A α :αє }.Then B {f -1 (A α ):αє }. By hypothess, there exsts a fnte subset of such that f(b) {A α : α }.Then B { f -1 (A α ):α }.By hypothess, there exsts a fnte subset of such that B {f -1 (A α ):α }.Therefore, we have f(b) {A α :α } whch shows that f(b) s wgrα-compact relatve to Y. Theorem: 4.7 If f: X Y s prewgrα-open bjecton and Y s wgrα-compact space, then X s a wgrα-compact space. 860 Page

5 Let {U α :α Internatonal Journal of Modern Engneerng Research (IJMER) Vol.3, Issue.2, March-Aprl pp ISSN: } be a wgrα-open cover of X.So X= U and then Y=f(X) =f( U ) = f ( ).Snce f s prewgrαopen, for each α,f(u α ) s wgrα-open set. By hypothess, there exsts a fnte subset Therefore, X=f -1 (Y)=f -1 ( f ( U ) )= U Ths shows that X s wgrα-compact. U of such that Y= f ( ) Theorem: 4.8 If f:x Y s wgrα-rresolute bjecton and X s wgrα-compact space, then Y s a wgrα-compact space. Let {U α :α } be a wgrα-open cover of Y. So Y= U and then X=f -1 (Y)= f -1 ( U ) = f 1 ( U ).Snce f s wgrα-rresolute, t follows that for each α, f -1 (U α ) s wgrα-open set. By wgrα-compactness of X, there exsts a fnte subset of such that X= 1 f ( U ). Therefore, Y=f(X)=f( f 1 ( U ) )= U.Ths shows that Y s wgrα-compact. Theorem: 4.9 A wgrα-contnuous mage of a wgrα-compact space s compact. Let f: X Y be a wgrα-contnuous map from a wgrα-compact space X onto a topologcal space Y. let {A : } be an open cover of Y.Then {f -1 (A ): } s wgrα-open cover of X. Snce X s wgrα-compact,t has fnte subcover,say{f -1 (A 1 ),f - 1 (A 2 ),,f -1 (A n )}.Snce f s onto,{ A 1,A 2,,A n } and so Y s compact. Theorem: 4.10 A space X s wgrα-compact f and only f each famly of wgrα-closed subsets of X wth the fnte ntersecton property has a non-empty ntersecton. X s wgrα-compact and A s any collecton of wgrα-closed sets wth F.I.P. Let A ={F α :α } be an arbtrary collecton of wgrα-closed subsets of X wth F.I.P, so that { F : o } ϕ (1),we have to prove that the collecton A has nonempty ntersecton,that s, {F α :α } ϕ (2).Let us assume that the above condton does not hold and hence {F α :α }=ϕ.takng complements of both sdes,we get {F α C :α }=X (3).But each F α beng wgrα-closed,whch mples that F C α s wgrα-open and hence from (3),we conclude that C={F C α :α } s an wgrα open cover of X. Snce X s wgrα-compact, ths cover C has a fnte subcover. C ={ F : o } s also an open subcover. Therefore X= { F C : o }.Takng complement, we get ϕ= { F : o }whch s a contradcton of (1).Hence {F α :α } ϕ. Conversely, suppose any collecton of wgrα-closed sets wth F.I.P has a empty ntersecton.let C ={G α :α },where G α s a wgrα-open cover of X and hence X= {G α :α }.Takng complements, we have ϕ= {G C α :α }.But G C α s wgrαclosed. Therefore the class A of wgrα-closed subsets wth empty ntersecton.so that t does not have F.I.P. Hence there there exsts a fnte number of wgrα-closed sets G C such that o wth empty ntersecton. That s,{ G : o }=ϕ.takng complement, we have { G C : o }= X.Therefore C of X has an open subcover C*={ G : o }.Hence (X,τ) s compact. Theorem :4.11 If f: (X,τ ) (Y,σ) s a strongly wgrα-contnuous onto map, where (X,τ) s a compact space, then (Y,σ) s wgrα-compact. Let {A : } be a wgrα-open cover of (Y,σ).Snce f s strongly wgrα-contnuous,{f -1 (A : } s an open cover (X,τ).As (X,τ) s compact, t has a fnte subcover say,{f -1 (A 1 ),f -1 (A 2 ),, f -1 (A n )} and snce f s onto, {A 1,A 2,,A n } s a fnte subcover of (Y,σ) and therefore (Y,σ) s wgrα-compact. Theorem :4.12 If a map f: (X,τ) (Y,σ) s a perfectly wgrα-contnuous onto map, where (X,τ) s compact, then (Y,σ) s wgrα-compact. Snce every perfectly wgrα-contnuous functon s strongly wgrα-contnuous. Therefore by theorem 4.11, (Y,σ) s wgrαcompact. V. Wgrα-Connected Spaces Defnton: 5.1 A Space X s sad to be wgrα-connected f t cannot be wrtten as a dsjont unon of two non-empty wgrα-open sets. U 861 Page

6 Vol.3, Issue.2, March-Aprl pp ISSN: Defnton: 5.2 A subset of X s sad to be wgrα-connected f t s wgrα-connected as a subspace of X. Defnton: 5.3 A functon f: (X,τ) (Y,σ) s called contra wgrα-contnuous f f -1 (V) s wgrα-closed n (X,τ) for each open set V n (Y,σ). Theorem: 5.4 For a space X, the followng statements are equvalent ()X s wgrα-connected. ()X and υ are the only subsets of X whch are both wgrα-open and wgrα-closed. ()Each wgrα-contnuous map of X nto some dscrete space Y wth atleast two ponts s a constant map. () () Let X be wgrα-connected. Let A be wgrα-open and wgrα-closed subset of X. Snce X s the dsjont unon of the wgrα-open sets A and A C, one of these sets must be empty. That s, A =υ or A=X. () () Let X be not wgrα-connected, whch mples X=A B,where A and B are dsjont non-empty wgrα-open subsets of X. Then A s both wgrα-open and wgrα-closed. By assumpton A=υ or A=X, therefore X s wgrα-connected. ()() Let f:x Y be wgrα contnuous map from X nto dscrete space Y wth atleast two ponts, then{ f -1 (y):yϵ Y} s a cover of X by wgrα-open and wgrα-closed sets. By assumpton, f -1 (y)=υ or X for each yϵ Y. If f -1 (y)=υ for all yϵ Y, then f s not a map. So there exsts a exactly one pont yϵ Y such that f -1 (y) φ and hence f -1 (y) =X. Ths shows that f s a constant map. ()() Let O υ be both an wgrα-open and wgrα-closed subset of X.Let f:x Y be wgrα-contnuous map defned by f(o)={y} and f(o C )={ω} for some dstnct ponts y and ω n Y. By assumpton f s constant, therefore O=X. Theorem: 5.5 Let f: X Y be a map: () If X s wgrα-connected and f s wgrα-contnuous surjectve, then Y s connected. () If X s wgrα-connected and f s wgrα-rresolute surjectve, then Y s wgrα-connected. () If Y s not connected, then Y= A B, where A and B are dsjont non-empty open subsets of Y. Snce f s wgrαcontnuous surjectve, therefore X=f -1 (A) f - 1 (B), where f -1 (A) and f -1 (B) are dsjont non-empty wgrα-open subsets of X. Ths contradcts the fact that X s wgrα-connected. Hence, Y s connected. ()Suppose that Y s not wgrα-connected, then Y=A B, where A and B are dsjont non-empty wgrα-open subsets of Y. Snce f s wgrα-rresolute surjectve, therefore X=f -1 (A) f -1 (B),where f -1 (A),f -1 (B) are dsjont non-empty wgrα-open subsets of X. So X s not wgrα-connected, a contradcton. Theorem: 5.6 A contra wgrα-contnuous mage of a wgrα-connected space s connected. Let f: (X,τ) (Y,σ) be a contra wgrα-contnuous from a wgrα-connected space X onto a space Y. Assume Y s not connected. Then Y=A B, where A and B are non-empty closed sets n Y wth A B=υ. Snce f s contra wgrαcontnuous,we have that f -1 (A) and f -1 (B) are non-empty wgrα-open sets n X wth f -1 (A) f -1 (B)= f -1 (A B) =f -1 (Y)=X and f -1 (A) f -1 (B)=f -1 (A B)= f -1 (υ).ths means that X s not wgrα-connected, whch s a contradcton. Ths proves the theorem. Theorem: 5.7 Every wgrα-connected space s connected. Let X be an wgrα-connected space. Suppose X s not connected. Then there exsts a proper non-empty subset B of X whch s both open and closed n X. Snce every closed set s wgrα-closed,b s a proper non-empty subsets of X whch s both wgrα-open and wgrα-closed n X. Therefore X s not wgrα-connected. Ths proves the theorem. Remark: 5.8 Converse of the above theorem need not be true as seen n the followng example. Example: 5.9 Let X={a,b,c},τ={υ,{a,b},X}.{X,τ} s connected. But {a} and {b} are both wgrα-closed and wgrα-open, X s not wgrαconnected. Theorem: 5.10 Let X be a T wgrα -space. Then X s wgrα-connected f X s connected. Suppose X s not wgrα-connected. Then there exsts a proper non-empty subset B of X whch s both wgrα-open and wgrαclosed n X. Snce X s T wgrα -space, B s both open and closed n X and hence X s not connected. Theorem: 5.11 Suppose X s wgrα-t 1\2 space. Then X s wgrα-connected f and only f X s gα-connected Suppose X s wgrα-connected. X s gα-connected. 862 Page

7 Vol.3, Issue.2, March-Aprl pp ISSN: Conversely, we assume that X s gα-connected. Suppose X s not wgrα-connected. Then there exsts a proper non-empty subset B of X whch s both wgrα-open and wgrα-closed n X.Snce X s wgrα-t 1\2 -space s both α-open and α-closed n X. Snce α-closed set s gα-closed n X, B s not gα-connected n X, whch s a contradcton. Therefore X s wgrα-connected. Theorem: 5.12 In a topologcal space (X,τ) wth at least two ponts, f αo(x,τ)=αc(x,τ),then X s not wgrα-connected. By hypothess, we have αo(x,τ)=αc(x,τ) and by the result,we have every α-closed set s wgrα-closed, there exsts some non-empty proper subset of X whch s both wgrα-open and wgrα-closed n X. So by theorem 5.4, we have X s not wgrαconnected. References [1] A.M. Al-Shban, rg-compact spaces and rg-connected spaces, Mathematca Pannonca17/1 (2006), [2] S.S Benchall, Pryanka M. Bansal, gb-compactness and gb-connectedness Topologcal Spaces, Int.J. Contemp. Math. Scences, vol. 6, 2011, no.10, [3] R. Dev, K. Balachandran and H.Mak, On Generalzed α-contnuous maps, Far. East J. Math., 16(1995), [4] S. Eswaran, A. Pushpalatha, τ*-generalzed Compact Spaces and τ*- Generalzed Connected Spaces n Topologcal Spaces, Internatonal Journal of Engneerng Scence and Technology, Vol. 2(5), 2010, [5] Y.Gnanambal and K.Balachandran, On gpr-contnuous Functons n Topologcal spaces, Indan J.Pure appl.math, 30(6), , June [6] A. Jayalakshm and C. Janak, wgrα- closed sets n Topologcal spaces, Int.Journal of Math. Archeve, 3(6), [7] A. Jayalakshm and C. Janak, wgrα-closed and wgrα-open Maps n Topologcal Spaces (submtted). [8] Levne.N, Strong Contnuty n Topologcal Spaces, Am Math. Monthly 1960; 67:267. [9] H. Mak, R. Dev and K.Balachandran, Generalzed α-closed sets n Topology, Bull. Fukuoka Unv. Ed. Part -III, 42(1993), [10] H.Mak, R.Dev and K.Balachandran, Assocated Topologes of Generalzed α-closed Sets and α-generalzed Closed Sets Mem.Fac. Sc. Koch.Unv. Ser.A. Math.15 (1994), [11] T.Nor, A Generalzaton of Perfect Functons, J.London Math.Soc., 17(2)(1978) [12] Nor.T,On δ-contnuous Functons, J.Korean Math.Soc 1980;16: [13] T.Nor,Super Contnuty and Strong Forms of Contnuty,Indan J.Pure Appl.Math.15 (1984),no.3, [14] A.Vadvel and K. Varamanckam, rgα-closed Sets and rgα-open Sets n Topologcal Spaces, Int. Journal of Math. Analyss, Vol. 3, 2009, no.37, Page