The Alexandrov-Urysohn Compactness On Single

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1 EID JFRI I. ROCKIRNI J. MRTIN JENCY 3 College of Vestsjaellad outhherrestraede 400 lagelse Demark. 3 Departmet of Mathematcs Nrmala College for wome Combatore Tamladu Ida. 3 E-mal: martajecy@gmal.com The lexadrov-urysoh Compactess O gle Valued Neutrosophc Cetered ystems bstract I ths paper we preset the oto of the sgle valued eutrosophc maxmal compact exteso sgle valued eutrosophc cetered system. Moreover the cocept of sgle valued eutrosophc absolute s appled to establsh the lexadrov -Urysoh compactess crtero. ome of the basc propertes are characterzed. Keywords gle valued eutrosophc cetered system sgle valued homeomorphsm sgle valued eutrosophc θ cotuous fuctos. eutrosophc θ. Itroducto Floret maradache [9] combed the o- stadard aalyss wth a tr compoet logc/set probablty theory wth phlosophy ad proposed the term eutrosophc whch meas kowledge of eutral thoughts. Ths eutral represets the ma dstcto betwee fuzzy ad tutostc fuzzy logc set. I 998 Floret maradache [6] defed the sgle valued eutrosophc set volvg the cocept of stadard aalyss. toe [0 ] appled the apparatus of Boolea rgs to vestgate spaces more geeral tha completely regular oes related to some extet to the fucto-theoretc approach. Usg these methods toe [0 ] obtaed a umber of mportat results o Hausdorff spaces ad fact troduced the mportat topologcal costructo that was later called the absolute. The frst proof of lexadrov-urysoh compactess crtero wthout ay axom of coutablty was gve by toe [0 ].Cech exteso topologcal spaces ad lexadrov-urysoh compactess crtero were costructed by Ilads ad Fom[7]. 345

2 Floret maradache urapat Pramak (Edtors) I ths paper the cocept of absolute sgle valued eutrosophc structure space ad the sgle valued eutrosophc maxmal compact exteso β() (sgle valued eutrosophc cech exteso) of a arbtrary sgle valued eutrosophc completely regular space s troduced. Further the lexadrov -Urysoh compactess crtero o sgle valued eutrosophc structure has bee studed.. Prelmares Defto.: [6] Let X be a space of pots (objects) wth a geerc elemet X deoted by x. sgle valued eutrosophc set (VN) X s characterzed by truth-membershp fucto T determacymembershp fucto I ad falsty-membershp fucto F. For each pot x X T(x) I(x) F(x) [0]. Whe X s cotuous a VN ca be wrtte as T( x) I ( x) F( x) / x x X. X Whe X s dscrete a VN ca be wrtte as T( x ) I( x ) F( x ) / x x X Defto.: [3] Let X be a o- empty set ad a collecto of all sgle valued eutrosophc sets of X. sgle valued eutrosophc structure o s a collecto of subsets of havg the followg propertes. φ ad are.. The uo of the elemets of ay sub-collecto of s. 3. The tersecto of the elemets of ay fte sub-collecto of s. The collecto together wth the structure s called sgle valued eutrosophc structure space. The members of are called sgle valued eutrosophc ope sets. The complemet of sgle valued eutrosophc ope set s sad to be a sgle valued eutrosophc closed set. Example.3: [3] Let X { a b} a b 3 4 where

3 a b a b a b a b Here ( ) s a structure space. Defto.4: [3] Let be a member of. sgle valued eutrosophc ope set U ( ) s sad to be a sgle valued eutrosophc ope eghborhood of f G U for some sgle valued eutrosophc ope set G ( ). Example.5: [3] Let X { a b} a b 3 4 where a b a b a b a b Let a b Here 4. 4 s the sgle valued eutrosophc ope eghbourhood of. Defto.6: [3] Let ( ) be a sgle valued eutrosophc structure space ad x T I F be a sgle valued eutrosophc set X. The the sgle valued eutrosophc closure of (brefly V N cl()) ad sgle valued eutrosophc teror of (brefly VN t()) are respectvely defed by VN cl() = {K: K s a sgle valued eutrosophc closed sets ad K} VN t() = {G: G s a sgle valued eutrosophc ope sets ad G }. 347

4 Floret maradache urapat Pramak (Edtors) Example.7: [3] Let X { a b} a b 3 4 where a b a b a b a b c a b a b c a b c a b 3 c Let a b. The VN t( ) { 3} c VN cl( ) { }. 4 Defto.8: [3] The ordered par ( ) s called a sgle valued eutrosophc Hausdorff space f for each par of dsjot members of there exst dsjot sgle valued eutrosophc ope sets U ad U such that U ad U. Example.9: [3] Let X { a b} a 0 b 0 where 3 a b a b a b Let a b a b Here ad are dsjot members of ad are dsjot sgle valued eutrosophc ope sets such that ad. Hece the ordered par ( ) s a sgle valued eutrosophc Hausdorff space. 348

5 Defto.0: [3] Let ( ) ad ( ) be ay two sgle valued eutrosophc structure spaces ad let f :( ) ( ) be a fucto. The f s sad to be sgle valued eutrosophc cotuous ff the pre mage of each sgle valued eutrosophc ope set ( ) s a sgle valued eutrosophc ope set ( ). Defto.: [3] Let ( ) ad ( ) be ay two sgle valued eutrosophc structure spaces ad let f :( ) ( ) be a bjectve fucto. If both the fuctos f ad the verse fucto f :( ) ( valued eutrosophc homeomorphsm. Defto.: [4] ) are sgle valued eutrosophc cotuous the f s called sgle Let f be a fucto from a sgle valued eutrosophc structure space ( ) to a sgle valued eutrosophc structure space ( ) wth f ) f ( ) where ( ) ad ( ( ).The f s called a sgle valued eutrosophc cotuous at f for every eghbourhood O of there exsts a eghbourhood O of such that f VN cl( O )) VN cl( O ).The fucto s called sgle valued eutrosophc ( cotuous f t s sgle valued eutrosophc cotuous at every member of. Defto.3: [3] fucto s called a sgle valued eutrosophc θ homeomorphsm f t s sgle valued eutrosophc oe to oe ad sgle valued eutrosophc θ cotuous both drectos. Defto.4: [3] Let ( ) be a sgle valued eutrosophc Hausdorff space. system p = {Uα : α = 3...} of sgle valued eutrosophc ope sets s called a sgle valued eutrosophc cetered system f ay fte collecto of the sets of the system has a o- empty tersecto. Example.5: [3] Let X { a b} a 0 b 0 where 3 349

6 Floret maradache urapat Pramak (Edtors) a b a b a b Let a b a b.let us cosder the system 0.50 p 3. p s a sgle valued eutrosophc cetered system sce has a o -empty tersecto. Let p { : } s also a sgle valued eutrosophc cetered system. Here p s a maxmal sgle valued eutrosophc cetered system. Defto.6: [3] The sgle valued eutrosophc cetered system p s called a maxmal sgle valued eutrosophc cetered system or a sgle valued eutrosophc ed f t caot be cluded ay larger sgle valued eutrosophc cetered system of sgle valued eutrosophc ope sets. Defto.7: [3] subset of a sgle valued eutrosophc structure space ( ) s sad to be a everywhere sgle valued eutrosophc dese subset ( ) f V N cl() =. Defto.8: [3] subset of a sgle valued eutrosophc structure space ( ) s sad to be a owhere sgle valued eutrosophc dese subset ( ) f X \ c s everywhere sgle valued eutrosophc dese subset. 3. gle valued eutrosophc Cech exteso Defto 3.: sgle valued eutrosophc cetered system p = {Uα} of sgle valued eutrosophc ope sets of s called a sgle valued eutrosophc completely regular system f for ay Uα p there exsts a Vα p ad a sgle valued eutrosophc cotuous fucto f o such that f () = for \Uα f () = 0 for Vα ad 0 f () for ay. I ths case Vα. s a sgle valued eutrosophc completely regularly cotaed Uα. 350

7 Defto 3.: sgle valued eutrosophc completely regular system s called a sgle valued eutrosophc completely regular ed f t s ot cotaed ay larger sgle valued eutrosophc completely regular system. Defto 3.3: Let ( ) ad ( ) be ay two sgle valued eutrosophc structure spaces. fucto f : ( ) ( ) of a sgle valued eutrosophc structure space ( ) oto a sgle valued eutrosophc structure space ( ) s a quotet fucto (or atural fucto) f wheever V s a sgle valued eutrosophc a sgle valued eutrosophc Note 3.4: ope set ( ) ad coversely. ope set ( ) f ( V ) s The maxmal sgle valued eutrosophc cetered systems of sgle valued eutrosophc ope sets ( sgle valued eutrosophc eds) regarded as elemets of the sgle valued eutrosophc space θ() fall to two classes: those sgle valued eutrosophc eds each of whch cotas all the sgle valued eutrosophc ope eghbourhoods of oe (obvously oly oe) member of ad the sgle valued eutrosophc eds ot cotag such systems of sgle valued eutrosophc ope eghbourhoods. The sgle valued eutrosophc eds of the frst type ca be regarded as represetg the members of the orgal sgle valued eutrosophc space ad those of the secod type as correspodg to holes. Defto 3.5: The collecto of all sgle valued eutrosophc eds of the frst type θ() s a sgle valued eutrosophc completely regular space ad t s also called the sgle valued eutrosophc absolute of whch s deoted by w(). I w() each member V s represeted by sgle valued eutrosophc eds cotag all sgle valued eutrosophc ope eghbourhoods of. It s obvous that w( ) B( V ) where B (V ) are the sgle valued eutrosophc eds p of that cota all the sgle valued eutrosophc ope eghbourhoods of V.The subset w() s mapped a atural way oto.if p w() the by defto ( p) V where V s the member whose sgle V 35

8 Floret maradache urapat Pramak (Edtors) valued eutrosophc ope eghbourhoods all belog to p ad s the atural fucto of w() oto. Lemma 3.6: sgle valued eutrosophc cetered system {Uα} of all sgle valued eutrosophc ope eghbourhoods of a member a sgle valued eutrosophc completely regular space s a sgle valued eutrosophc completely regular ed. Here {Uα} s a sgle valued eutrosophc completely regular cetered system.the Lemma wll be proved f t s possble to show that {Uα } s ot cotaed ay other sgle valued eutrosophc completely regular system. s a cotrary suppose that {Vα } s a sgle valued eutrosophc cetered completely regular system cotag {Uα } wth V { U }. ce V meets every sgle valued eutrosophc ope eghbourhood of VN cl( V )\ V.Let V be a elemet of {Vα} such that VN cl( V ) V.But the V N cl( V ).It follows that V does ot meet ay of the sgle valued eutrosophc ope eghbourhoods of so {Vα } caot be a sgle valued eutrosophc cetered system cotag {Uα }. Now we costruct a sgle valued eutrosophc structure space whch s deoted by ( ).Itsmembers are all sgle valued eutrosophc completely regular eds of ad ts sgle valued eutrosophc topology s defed as follows: Choose a arbtrary sgle valued eutrosophc ope set U ad the collectoo of all sgle valued eutrosophc cetered completely regular eds of that cota U as a member s to be a sgle valued eutrosophc ope eghbourhood of each of them. Lemma 3.7: sgle valued eutrosophc completely regular ed p = {Uα} of a sgle valued eutrosophc structure space has the followg propertes:. If Uβ Uα p the Uβ p.. The tersecto of ay fte umber of members of p belogs to p. U 35

9 sserto () s obvous. () Let U U... U p U U... U p ad f f... f be fuctos such that f ( ) 0 o VN cl( U ) f ( ) o \ U. The the fucto f ) f ( )... f ( ) s zero o VN cl( U )... VN cl( U ) ad a fortor o ( VN cl(( U )... ( U )). ce \ ( U ) U \ \ the f ( )... f ( ) at each member of ( U ). Puttg f ( ) wheever f ( )... f ( ) ad f ( ) f( )... f ( ). Whe ths sum s less tha t may be obtaed a fucto f () such that 0 f ( ) f ( ) 0 o VN cl( U ) ad f ( ) o ( U ). Hece the sgle valued eutrosophc system p must cota t would ot be maxmal sgle valued eutrosophc completely regular system. Corollary 3.8: O U U ad U \ otherwse OV OU V. For f p O U OV the U p ad V p. By Lemma 3.7 U V p that s p.therefore O O O. If p OU V the U V p ad by the same O U V Lemma U p ad V p. Therefore O O O U V U V.Thus OU OV OU V. Lemma 3.9: U V U V p OU ad OV p. That s p O U OV.Hece sgle valued eutrosophc structure space α () s a sgle valued eutrosophc Hausdorff exteso of. The sgle valued eutrosophc structure space α () s a sgle valued eutrosophc Hausdorff space. Let p ad q be ay two dsjot members of α (). The t s easy to fd U p ad V p such that U V = φ for otherwse the sgle valued eutrosophc cetered system cosstg of all the members of p ad all the members of q would be a sgle valued eutrosophc cetered completely regular system cotag p ad q whch s mpossble. O U ad O V assocated wth ths U ad V are dsjot sgle valued eutrosophc ope eghbourhoods of p ad q α (). 353

10 Floret maradache urapat Pramak (Edtors) 354 It shall be detfed that the member wth the sgle valued eutrosophc ed = {Uα} cosstg of all the sgle valued eutrosophc ope eghbourhood of. The OU = U whch shows that s sgle valued eutrosophc topologcally embedded α () ad sce t s easy to see that s everywhere sgle valued eutrosophc dese α (). Therefore α () s a sgle valued eutrosophc exteso of. Ths proves the Lemma. Note 3.0: I a sgle valued eutrosophc completely regular space the sgle valued eutrosophc caocal eghbourhood forms a base. Lemma 3.: sgle valued eutrosophc structure space α ()ca be cotuously mapped oto every sgle valued eutrosophc compact exteso of such a way that the members of rema fxed. Let b() be ay sgle valued eutrosophc compact exteso of. Each member b() determes the sgle valued eutrosophc cetered system p p = {Uα} cosstg of all sgle valued eutrosophc ope eghbourhoods of b().by Lemma 3.6 ths s a sgle valued eutrosophc completely regular system ad a sgle valued eutrosophc maxmal.the sgle valued eutrosophc cetered system q = {Vα = Uα } s a sgle valued eutrosophc completely regular system.if d = {Hα } α () cotas a sgle valued eutrosophc cetered system q the we defe φ o α () as φ(d) = q.ce b() s a sgle valued eutrosophc Hausdorff exteso d ca cota oly oe such sgle valued eutrosophc cetered system q.hece the fucto φ s well-defed.ce a arbtrary sgle valued eutrosophc completely regular system ca be exteded to a sgle valued eutrosophc completely regular ed φ s oto. It s easy to see that f the φ() =.φ s defed o the whole of α ().For f d = {Hα} α b( ) () the VN cl( H ) (because b() s a sgle valued eutrosophc compact space).let VN cl( H ) b( ).The the sgle valued

11 eutrosophc cetered system d q cosstg of all members Hα d ad all members Uα q s a sgle valued eutrosophc completely regular ad sce d s a sgle valued eutrosophc maxmal completely regular system d q = d that s q d so that φ(d) =. Let b() ad Uα be ay sgle valued eutrosophc ope eghbourhood of b(). ssumg Uα s a caocal sgle valued eutrosophc ope eghbourhood. Put Vα = Uα. Let d α () ad φ(d)=. The O V s a sgle valued eutrosophc ope eghbourhood of d α ().To show that φ( O ) V N cl(uα ).For ths t s clear that Vα q f ad oly f V Uα. Now f d O V the Vα d.if φ(d ) = / V N cl(uα ) the some sgle valued eutrosophc ope eghbourhood of whch does ot cotaed Uα but the Vα / q so that Vα / d that s d / OVα Ths cotradcts our assumpto.ce b() s sgle valued eutrosophc regular space φ s a sgle valued eutrosophc cotuous ad the Lemma s proved.the sgle valued eutrosophc set set of forms a base α (). Lemma 3.: O U where U s a caocal sgle valued eutrosophc ope The sgle valued eutrosophc structure space α () s a sgle valued eutrosophc completely regular space. Let p ={Uα } α () ad let U be ay sgle valued eutrosophc completely regular cotaed the caocal sgle valued eutrosophc ope set U.ssume that VN cl( O ) (α ()\ O ).The there s a member q = {VN cl(vα )} U U such that q V N cl( O ) (α ()\ U O ).The relato q VN cl( O ) meas U U that every Vα q meets U ad the relato q α ()\ O equvalet to q / meas that every Vα meets \U.ce U s a caocal sgle valued eutrosophc ope set t follows that Vα meets \VN cl(u ). If V ad V are sgle valued eutrosophc ope sets such that V s sgle valued eutrosophc completely regularly cotaed \VN cl(v) ad V = V V q U O U 355

12 Floret maradache urapat Pramak (Edtors) the ether V q or V q. Now let f (B) be a fucto that s zero o V N cl( U ) ad o \U.uch a fucto exsts sce U s sgle valued eutrosophc completely regularly cotaed U.lso let 0 < a < b < ad let Γ(a b) be the sgle valued eutrosophc ope set {B : a < f (B) < b}.by assumpto every Vα q has o- empty tersecto wth Γ(a b).for otherwse Vα splts to two sgle valued eutrosophc ope sets Vα ad Vα such that Vα s sgle valued eutrosophc completely regularly cotaed \V N cl(vα ) ad V α U = φ Vα (\VN cl(u )) = φ. The last equato cotradcts the fact that q V N cl ( O ) (α ()\ O ). U Cosder the sgle valued eutrosophc ope sets Γ(a b) where 0 < a < a0 < b0 < b < ad a0 ad b0 are fxed.they form a sgle valued eutrosophc completely regular system whch must be cotaed q.but Γ(a b) U that s Γ(a b) Γ(\V N cl(u)) = φ ad hece q /(α ()\ O ). Ths cotradcto shows that V N cl U ( O ) U O U from whch t follows that α () s a sgle valued eutrosophc regular space. To prove that t s a sgle valued eutrosophc completely regular space. Let Γt 0 t deote the set of all B for whch f (B) < t. It has show that f t < t the VN cl( O t ) O t.hece t follows that U O U s sgle valued eutrosophc completely regularly cotaed O. U 356 Lemma 3.3: The sgle valued eutrosophc structure space α () s a sgle valued eutrosophc compact. If H s a sgle valued eutrosophc ope set of α () the there exsts a sgle valued eutrosophc ope set U (H ) of such that H O VN cl( ) the U (H ) U ( H ) H = α Uα.If H s sgle valued eutrosophc completely regularly embedded G the O s clearly sgle valued eutrosophc completely regularly embedded U (H ) O U (G).uppose that α () s ot a sgle valued eutrosophc compact space. The by

13 Tychooff s theorem there e x s t s a sgle valued eutrosophc completely r e gu l a r s p a c e α () ξ cotag α () as a everywhere sgle valued eutrosophc dese set. Let Hα be the set of all sgle valued eutrosophc ope sets of α () for whch Hα ξ s a sgle valued eutrosophc ope eghbourhood of ξ α () ξ. The {Hα} s a sgle valued eutrosophc completely regular system α (). Hece O U H ) s also a sgle valued eutrosophc completely regular space. T h u s O U H ) = φ. c e α () s a sgle valued eutrosophc cetered system p = { (Hα)} of sgle valued eutrosophc completely regular too. But p OU H ) for every α Λ that s O α U H ) = φ. Ths cotradcto proves the lemma. Proposto 3.4: For ay sgle valued eutrosophc completely regular space the sgle valued eutrosophc structure space α () cocdes wth the Cech exteso β() upto a sgle valued eutrosophc homeomorphsm leavg the members of fxed. The proof follows mmedately from Lemma 3. ad Lemma 3.3 ad the uqueess of a maxmal sgle valued eutrosophc compact exteso. 4. The lexadrov - Urysoh compactess I ths secto the cocept of sgle valued eutrosophc absolute s appled to establsh the lexadrov - Urysoh compactess. Property 4.: ( ( ( ( If F F F... F 3 wth F o -empty the o-empty so s F ~ ). F ~ ( partcular f F s Let B F ad let q = {G } be a sgle valued eutrosophc ed of F cotag a sgle valued eutrosophc cetered system of sgle valued eutrosophc ope sets G F such that B V N t(v N cl(g )).It may be assumed that t has bee costructed systems q = {G } of F such that q cotas all the sgle valued 357

14 Floret maradache urapat Pramak (Edtors) eutrosophc ope sets G F for whch B V N t(v N cl(g )) ad all sgle valued eutrosophc ope sets whose tersecto wth F s some G. Now costruct q +. By defto q + cossts of all sets G + F+ for whch B V N t(v N cl(g + )) ad of all sgle valued eutrosophc ope sets whose tersecto wth F s some G.It s easy to show that q + s a sgle valued eutrosophc cetered system. Thus for each costruct a sgle valued eutrosophc cetered system q. Let p = {H}deote the sgle valued eutrosophc ed of cotag q.to show that p F ~.It follows from the costructo of p that f H F q for some ad some sgle valued eutrosophc ope set H the H p.to ~ show that p F.Let H be a sgle valued eutrosophc ope set of such that B V N t(v N cl(h F )).The H F q ~ ad hece H p that s p F.Hece the proof. Property 4.: If F s a sgle valued eutrosophc H closed the F ~ s sgle valued eutrosophc compact (ad hece sgle valued eutrosophc closed θ()). Let {Hα} be ay sgle valued eutrosophc coverg of F ~ by sgle valued eutrosophc ope sets F ~.They may be exteded to sgle valued eutrosophc ope w(). It may assume that each of that each of the exteded sets has the form O U where U s a sgle valued eutrosophc ope set. Otherwse {Hα } may be replaced by a fer coverg for whch ths codto holds.o t may be assumed that {Hα } s a sgle valued eutrosophc coverg of F by sets sgle valued eutrosophc ope w() of the form O where U U α s sgle valued eutrosophc ope. Let B F. Le t B H deote the uo of a fte umber of sgle valued eutrosophc ope sets Hα coverg the sgle valued eutrosophc compact set ( B ).It s clear that f or m s B H h as t he B O U B where U s a s gle va lue d eutrosophc ope set ad s maxmal amog the sgle valued eutrosophc ope sets H for whch O H O B U.From the above 358

15 t follows that the sgle valued eutrosophc cetered system VN t( U B F) s a sgle valued eutrosophc coverg of F.ce F s sgle valued eutrosophc H closed choose a fte umber of members of ths sgle valued eutrosophc coverg such that B VN cl( VN t( VN cl( U F))) F where the closure s take F both ~ cases. To show that O B F.ce the uo O U U B has the property that U B VN t( VN cl( F U)) for ay Bthe a arbtrary sgle valued eutrosophc ~ ed p F cotas U ad hece belogs to some O. Thus for oly those H U B α that make up O U B ad take ther tersectos wth F ~ t h e r e q u r e d fte coverg s obtaed. Defto 4.3: sgle valued eutrosophc Hausdorff space s a sgle valued eutrosophc compact space f ad oly f every (ot ecessarly coutable) well-ordered decreasg sequece of o-empty sgle valued eutrosophc closed sets has a o-empty tersecto. Theorem 4.4: (lexadrov - Urysoh compactess) sgle valued eutrosophc Hausdorff space s a sgle valued eutrosophc compact space f ad oly f each of ts sgle valued eutrosophc closed subset s sgle valued eutrosophc H closed. Necessty: The ecessty of ths codto follows from Property 4.. ce a sgle valued eutrosophc compact space every sgle valued eutrosophc closed subset s a sgle valued eutrosophc compact space ad hece sgle valued eutrosophc H closed. uffcecy: Let be a sgle valued eutrosophc Hausdorff space w() be ts sgle valued eutrosophc absolute ad be a sgle valued eutrosophc atural fucto of w() oto. lso let F be ay sgle valued eutrosophc subset of. It ca be assocated t wth a certa sgle valued eutrosophc subset F ~ of w() defed by sayg that the member p s ( B) B belogs to F ~ f p O for every U satsfyg the U codto B V N t(v N cl(u F )).By the costructo of F ~ t s cotaed 359

16 Floret maradache urapat Pramak (Edtors) the complete sgle valued eutrosophc verse mage ( F ) of F w(). F ~ s called as the reduced verse mage of F w(). The proof of the lexadrov - Urysoh compactess sgle valued eutrosophc topology s based o the propertes dscussed above. For suppose that the codtos of the theorem are satsfed ad that {Fα} s a wellordered decreasg system of sgle valued eutrosophc closed sets of. The by Property 4. the set F ~ f o r m a sgle valued eutrosophc cetered system w(). lso sce s all the F s are sgle valued eutrosophc compact space (Property 4.) hece Let C F ~. The πs (C) Fα for every α that s α Fα = φ as requred. F ~. Property 4.5: y well-ordered sequece of decreasg sgle valued eutrosophc H closed sets a sgle valued eutrosophc Hausdorff space has a o-empty tersecto. From the proof of Property 4. t s easy to see that ( F ~ ) F. However geeral F ~ does ot cocde wth ( F ). lso the proof of Theorem 4.4 t caot be take ( F ) s stead of F ~ sce the complete verse mage of a sgle valued eutrosophc H- closed set eed ot be sgle valued eutrosophc compact. I fact let be a sgle valued eutrosophc Hausdorff space ad F a sgle valued eutrosophc H closed subset such that there s a member \F for whch there does ot exst dsjot sgle valued eutrosophc ope eghbourhoods of ad F. Note that a sgle valued eutrosophc Hausdorff space two dsjot sgle valued eutrosophc compact sets have dsjot sgle valued eutrosophc ope eghbourhoods. If ( F ) were sgle valued eutrosophc compact the the sgle valued eutrosophc compact sets ( F ) ad ( ) would have dsjot sgle valued eutrosophc ope eghbourhoods w() say s U ad V. The t follows from the proof of the theorem that VN t(v N cl(π (U )) ad V N t(v N cl(π (V ))) would be dsjot sgle valued eutrosophc ope eghbourhoods of F ad whch cotradcts our assumpto. s s s 360

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