A new approach towards characterization of semicompactness of fuzzy topological space and its crisp subsets

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1 2013 (2013) 1-6 Avalable onlne at Volume 2013, Year 2013 Artcle ID jfsva-00133, 6 Pages do: /2013/jfsva Research Artcle A new approach towards characterzaton of semcompactness of fuzzy topologcal space and ts crsp subsets Pradp Kumar Gan 1, Ramkrshna Prasad Chakraborty 2, Madhumangal Pal 3 (1) Department of Mathematcs, Kharagpur College, Inda, Kharagpur, Paschm Mednpur , West Bengal, Inda. (2) Department of Mathematcs, Hjl College, Hjl, Kharagpur, Paschm Mednpur , West Bengal, Inda. (3) Department of Appled Mathematcs wth Oceanology and Computer Programmng, Vdyasagar Unversty, Mdnapore , West Bengal, Inda. Copyrght 2013 c Pradp Kumar Gan, Ramkrshna Prasad Chakraborty and Madhumangal Pal. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract It s wdely accepted that one of the most satsfactory generalzaton of the concept of compactness to fuzzy topologcal spaces s α compactness, frst ntroduced by Gantner et al. n 1978, followed by further nvestgatons by many others. Chakraborty et al. ntroduced fuzzy semcompact set and nvestgated and characterzed fuzzy semcompact spaces n terms of fuzzy nets and fuzzy pre f lterbases n In ths paper, we propose to ntroduce a new approach to characterze the noton of α semcompactness n terms of ordnary nets and f lters. Ths paper deals also wth the concept of α semlmt ponts of crsp subsets of a fuzzy topologcal space X and the concept of α semclosed sets n X and these concepts are used to defne and characterze α semcompact crsp subsets of X. Keywords: Fuzzy topologcal spaces, α semcompactness, α semcluster pont of nets and flters, α adherent pont, α semadherent pont, α lmt pont, α semlmt pont, α closure, α semclosure. 1 Introducton The dea of fuzzy set was orgnated from the classcal paper of L.A.Zadeh [18] n Subsequently many researchers have worked on varous basc concepts from general topology usng fuzzy sets and developed the theory of fuzzy topology. In recent years fuzzy topology has been found to be very useful n solvng many practcal problems. The notons of the sets and functons n fuzzy topologcal spaces are used extensvely n many engneerng problems, computatonal topology for geometrc desgn, computer-aded geometrc desgn, engneerng desgn research and mathematcal scences. El-Nasche [8, 9] has shown that the noton of fuzzy topology may be applcable to quantum physcs n connecton wth strng theory and e theory and fuzzy topology may be used to provde nformaton about the elementary partcles content of the standard model of hgh energy physcs. Shhong Du. et al. [5] are currently workng to fuzzfy the 9-ntersecton Egenhofer model [6, 7] for descrbng topologcal relatons n Geographc Informaton System(GIS) query. X.Tang [15] has used a slghtly changed verson of Chang s [2] fuzzy topologcal spaces to model spatal objects for GIS database and Structured Query Language(SQL) for GIS. In-depth analytcal study of fuzzy set theory s stll requred to provde more and more nformaton about any mathematcal system and ts subsystems to the modern scentfc research n the arena of mathematcal scences and physcal scences. The concept of compactness s one of the central and mportant concepts of paramount nterest to topologst and t seems to be the most celebrated type among all the coverng propertes. Its enormous use and potentalty for numerous applcatons nduced mathematcans to generalze Correspondng author. Emal address: mmpalvu@gmal.com Tel:

2 Page 2 of 6 the concept n fuzzy settng. It was Chang [2] who frst ntroduced the concept of compactness n fuzzy topologcal space. Ths attempt has been followed by Goguen [12] who after pontng out a drawback n the defnton of Chang, proved Alexander s subbase theorem, but could establsh Tychonoff theorem only for fnte products. Thereafter, Wong [17], Wess [16] and Lowen [13] treated compactness for fuzzy topologcal spaces n dfferent ways. But t s seen that all those approaches contan many lmtatons. In 1978, Gantner, Stenlage and Warren [10] dd a splendd work towards the process of fuzzfyng the concept of compactness and ntated a new defnton of fuzzy compactness, termed as α compactness, by whch they could assgn degrees to compactness and wth ths defnton they fnally generalzed Tychonoff theorem and even 1 pont compactfcaton. It seems from the subsequent nvestgatons that the theory of α compactness by Gantner et al. [10] s the most satsfactory one. Ths vew has also been endorsed by Malghan and Benchall [14] who also treated α compactness vs-à-vs α per f ect maps n fuzzy topologcal spaces and studed a verson of local α compactness weaker than that of Gantner et al. [10]. Georgou and Papadopoulos [11] dealt wth α compactness usng the noton of fuzzy upper lmt. In a very recent paper Chakraborty et al. [3] ntroduced fuzzy semcompact set and nvestgated and characterzed fuzzy semcompact spaces n terms of fuzzy nets and fuzzy pre f lterbases. In secton 3, we propose to defne and characterze α semcompactness n terms of ordnary nets and f lters. Ths seems to be qute new approach n as much as to our knowledge, n almost none of the theores concernng the nvestgatons of numerous fuzzy topologcal concepts, ordnary nets and f lters have been nvolved so far. In secton 4, we propose to ntroduce α semclosed sets and α semcompactness for crsp subsets of fuzzy topologcal spaces and wll carry on the nvestgaton of α semcompactness n a greater detal. At the same tme, we shall strve to ultmately acheve certan expected results n analogy to those known for semcompact sets vs-à-vs semclosed sets n a topologcal space. 2 Prelmnares Throughout ths paper, by (X,τ) or smply by X we mean a fuzzy topologcal space (henceforth abbrevated as fts.) n Chang s [2] sense and to denote A to be a fuzzy set n X, we shall sometmes wrte A I X, where I = [0,1]. For A,B I X, we wrte A B f A(x) B(x), for each x X. For a fuzzy set A n X, Cl(A), Int(A) and 1 A wll respectvely denote the closure, nteror and complement of A. For a famly ζ = {A α : α Λ} ( here and henceforth also, Λ denotes an ndexng set) of fuzzy sets n X, the unon A α and the ntersecton A α are fuzzy sets defned respectvely by ( A α )(x) = sup{a α (x) : α Λ} and ( A α )(x) = n f {A α (x) : α Λ}, for each x X[18]. 3 α Semcompact Fuzzy Topologcal Spaces We recall from [10] the defnton of α shadng of a fts. Defnton 3.1. Let X be a fts. A collecton ζ I X s called an α shadng of X, where 0 < α < 1, f for each x X there exsts some U x ζ such that U x (x) > α. A subcollecton ζ 0 of an α shadng ζ of X, that s also an α shadng of X, s called an α subshadng of ζ. Remark 3.1. It s clear from the above defnton that a collecton ζ of fuzzy sets n a fts. X s an α shadng ff sup{u(x) : U ζ } > α for each x X. Defnton 3.2. A fts. X s sad to be α semcompact f each α shadng of X by fuzzy semopen sets of X has a fnte α subshadng. The followng defnton s also gven n [10] for an L-fuzzy space, n slghtly dfferent forms. We prefer to ncorporate here, a complete proof of the sad theorem n our settng. Defnton 3.3. [10] A famly {F : Λ} of fuzzy sets n a fts. X s sad to have α fnte ntersecton property (to be abbrevated as α FIP) f for each fnte subset Λ 0 of Λ there s some x X such that n f {F (x) : Λ 0 } 1 α. Theorem 3.1. A fts. X s α semcompact ff for every famly ζ = {F : Λ} of fuzzy semclosed sets n X wth α FIP, there s some x X such that n f {F (x) : Λ} 1 α. Proof. Let X be α semcompact and let ζ = {F : Λ} be a famly of fuzzy semclosed sets n X wth α FIP. If possble, let for each x X, ( F )(x) < 1 α..e, { (1 F )}(x) > α, for each x X. Hence Ω = {1 F : Λ} s an α shadng of X by fuzzy semopen sets. By α semcompactness of X, there exsts a fnte subset Λ 0 of Λ such that,{ (1 F )}(x) > α, for each x X. 0.e,( F )(x) < 1 α, for each x X. Ths mples ζ does not have α FIP, a contradcton. Hence n f {F (x) : Λ} 1 α, 0 for each x X. Conversely, let ζ = {F : Λ} be an α shadng of X by fuzzy semopen sets. That s, ( F ))(x) > α. Then Ω = {1 F : Λ}

3 Page 3 of 6 s a famly of fuzzy semclosed sets such that { (1 F )}(x) < 1 α, for each x X. Then n vew of the hypothess, Ω cannot have α FIP. Thus for a fnte subset Λ 0 of Λ and for each x X, we have {1 ( F )(x)} = { (1 F )}(x) < 1 α, for each 0 0 x X. Ths mples ( F )(x) > α, for each x X. Hence X s α semcompact. 0 We now defne the concept of a sort of cluster ponts of ordnary nets and flters n a fts. and ultmately use them to characterze α semcompactness of a fts. Defnton 3.4. A pont x X s sad to be an α semcluster pont of an ordnary net {S n : n (D, )}, ((D, ) beng any drected set) n a fts. X f for each fuzzy semopen set U wth U(x) > α and for each m D, there exsts a k D such that k m n D and U(S k ) > α. Theorem 3.2. A fts. X s α semcompact ff every ordnary net n X has an α semcluster pont n X. Proof. Let X be α semcompact. If possble, let there be a net {x n : n (D, )} n X havng no α semcluster pont n X. Then for each x X, there exsts a fuzzy semopen set U x wth U x (x) > α and m x D such that U x (x n ) α, for all n m x, (n D). Then Ω = {U x : x X} s a collecton of fuzzy semopen sets such that for any fnte subcollecton {U x1,u x2,u x3,...,u xk } of Ω, there exsts m D such that m m x1,m x2,m x3,...,m xk and ( k U x )(x n ) α, for n m (n D). Ths mples ( k (1 U x ))(x n ) 1 α. Then the collecton Ω = {1 U x : x X} of fuzzy semclosed sets has α FIP. Then by theorem (3.1), there exsts y X such that ( (1 U x ))(y) 1 α, x X..e, ( U x )(y) α, x X..e, U x (y) α, for all U x Ω. In partcular, U y (y) α, x x contradctng the defnton of U y. Hence the gven net n X has an α semcluster pont n X. Conversely, let every net n X has an α semcluster pont n X. Let Ω = {U : Λ} be an arbtrary collecton of fuzzy semclosed sets n X wth α FIP. Let Λ denotes the collecton of all fnte subsets of Λ. Then (Λ, ) s a drected set, where µ,λ Λ, µ λ f µ λ. Let us put F µ = U for each µ Λ. As Ω has α FIP for each µ Λ, there s some x µ (say) n X such that µ nf{u (x µ ) : µ} 1 α. By vrtue of theorem (3.1), X wll be α semcompact f nf{u (z) : Λ} 1 α, for some z X. If possble let nf{u (z) : Λ} < 1 α, for each z X. Now {x µ : µ (Λ, )} s clearly a net n X and hence by hypothess, t has an α semcluster pont y X. Then nf{u (y) : Λ} < 1 α and hence there exsts 0 Λ such that U 0 (y) < 1 α. That s, (1 U 0 )(y) > α. Snce { 0 } Λ, there exsts µ 0 Λ wth µ 0 { 0 } (.e, 0 µ 0 ) such that (1 U 0 )(x µ0 ) > α. Then U 0 (x µ0 ) < 1 α. As 0 µ 0, nf{u (x µ0 )} {U 0 (x µ0 )} < 1 α, a contradcton. Hence nf{u (z) : Λ} 1 α, for some z X. Then by vrtue of theorem (3.1), X s α semcompact. Defnton 3.5. A pont x X, where X s a fts, s sad to be an α semcluster pont of a flter base ζ on X f for each fuzzy semopen set U wth U(x) > α and for each F ζ there exsts x F F such that U(x F ) > α. Theorem 3.3. A fts X s α semcompact ff every flterbase ζ on X has an α semcluster pont n X. Proof. Let be X be α semcompact. Let there exsts, f possble, a flterbase ζ on X havng no α semcluster pont n X. Then for each x X, there exsts a fuzzy semopen set U x wth U x (x) > α and there exsts F x ζ such that U x (y) α for each y F x. Thus Ω = {U x : x X} s an α shadng of X by fuzzy semopen sets of X. By α semcompactness of X, there exsts fntely many ponts x 1,x 2,x 3,...,x n X such that Ω 0 = {U x1,u x2,u x3,...,u xn } s agan an α-shadng of X. Now let F ζ such that F {F x1 F x2 F x3... F xn } [ ζ s flterbase]. Then U x (y) α for each y F and for = 1,2,...,n. Thus Ω 0 fals to be an α shadng of X, a contradcton. Hence every flterbase on X has an α semcluster pont. Conversely, let every flterbase ζ on X has an α semcluster pont n X. If possble let {y n : n (D, )} be a net n X havng no α semcluster pont n X. Then for each x X, there s a fuzzy semopen set U x wth U x (x) > α and there exsts some m x D such that U x (y n ) α for all n m x (n D). Thus B = {F x : x X}, where F x = {y n : n m x }, s a subbase for a flterbase ζ on X, where ζ conssts of all fnte ntersectons of members of B. By hypothess, ζ has an α semcluster pont z X. But there s a fuzzy semopen set U z wth U z (z) > α and there exsts m z D such that U z (y n ) α f or all n m z. That s, for all p F z, where F z B ζ, U z (p) α. Hence z cannot be an α semcluster pont of the flterbase ζ, a contradcton. Hence the net {y n : n (D, )} n X has an α semcluster pont. By theorem (3.2), α semcompactness of X follows. 4 α Semclosed sets, α Semcompact sets In ths secton we ntroduced a class of specal type of crsp subsets of X whch are defned by the fuzzy sets n a fts (X,τ). The ntroduced concept s developed to some extent, whch s used as a supportng tool for our ultmate am of ntroducng and studyng α semcompactness for crsp subsets along wth α semclosedness. We recall from [4] the followng defntons n order to ntroduce our proposed defnton of α semclosed sets.

4 Page 4 of 6 Defnton 4.1. Let (X,τ) be a fts and A X. An element x X s sad to be an α adherent (α lmt) pont of A (0 < α < 1) f for each fuzzy open set U n X wth U(x) > α, there exsts y A (y A \ {x}) such that U(y) > α. The set of all α lmt ponts of A s denoted by A α. Defnton 4.2. A subset C of X s sad to be α closed f t contans all ts α lmt ponts. Defnton 4.3. α-closure of a set A X, denoted by α-cla, s defned as α ClA = A A α. Remark 4.1. Obvously A α ClA and f A α A then α ClA A. A set A X s sad to be α closed ff α ClA = A. Now we defne α semclosed set by defnng α semadherent pont and α semlmt pont of a crsp subset A of a fts. (X,τ). Defnton 4.4. Let (X,τ) be a fts and A X. An element x X s sad to be an α semadherent (α semlmt) pont of A (0 < α < 1) f for each fuzzy semopen set U n X wth U(x) > α, there exsts y A (y A \ {x}) such that U(y) > α. The set of all α semlmt ponts of A wll be denoted by A α slp. Defnton 4.5. A subset C of X s sad to be α semclosed f t contans all ts α semlmt ponts. Defnton 4.6. α semclosure of a set A X, denoted by α SclA, s defned as α SclA = A A α sl p. Remark 4.2. Obvously A α SclA and f A α slp A then α SclA A. A set A X s sad to be α semclosed ff α SclA = A. Lemma 4.1. (a) A B A α slp B α slp and hence α SclA α SclB. (b) α Scl(A B) α SclA α SclB. Proof. (a) Let A X and let x A α slp. Ths mples x s an α semlmt pont of A. That s, for each fuzzy semopen set U n X wth U(x) > α, there exsts y A \ {x} such that U(y) > α. As A B, t follows that y B \ {x}. Therefore, x s an α semlmt pont of B. That s, x B α slp. Hence A α sl p B α sl p. Also α SclA = A A α sl p B B α sl p = α SclB. Hence α SclA α SclB. (b) Proof s obvous Defnton 4.7. [1] Let (X,τ) and (Y,τ 1 ) be two fts. A functon f : (X,τ) (Y,τ 1 ) s sad to be fuzzy semcontnuous f for every fuzzy open set V n τ 1, f 1 (V ) s fuzzy semopen set n X. Theorem 4.1. Let (X,τ) and (Y,τ 1 ) be two fts. If a functon f : (X,τ) (Y,τ 1 ) s fuzzy semcontnuous then f 1 (C) s a α semclosed set n X for every α closed set C n Y. Proof. Let C be a α closed set n Y. Let x X \ f 1 (C). Then f (x) / C [x / f 1 C]. As C s a α closed set n Y, f (x) s not a α lmt pont of C. Then there exsts V τ 1, such that V ( f (x)) > α and V (y) α for all y C [ actually for all y C \ { f (x)}) ]. Snce f s fuzzy semcontnuous, f 1 (V ) s fuzzy semopen set n X. Let z f 1 (C). Then f (z) C. Now f 1 (V )(x) = V ( f (x)) > α and f 1 (V )(z) = V ( f (z)) α for all z f 1 (C). Thus x s not a α semlmt pont of f 1 (C). Hence f 1 (C) s α semclosed set n X. Defnton 4.8. Let (X,τ) and (Y,τ 1 ) be two fts. A functon f : (X,τ) (Y,τ 1 ) s sad to be fuzzy semopen map f for each fuzzy open set A n X, f (A) s fuzzy semopen set n Y. Theorem 4.2. If f : (X,τ) (Y,τ 1 ) be a bjectve fuzzy semopen map, then the mage of a α closed set n X s a α semclosed set n Y. Proof. Let C X be a α closed set n X. Let y Y \ f (C). Then there exsts a unque z X such that f(z) = y [ snce f s bjectve]. As y / f (C), z / C. Now as C s α closed set n X, there exsts fuzzy open set V n τ such that V (z) > α and V (p) α for all p C [actually for all p C \ {z} ]. As f s fuzzy semopen map, we have f (V ) s fuzzy semopen set n Y. Now (V )(y) = V (z) > α. That s, f (V )(y) > α. Let t f (C). Then there exsts t 0 C such that f (t 0 ) = t. As f (V )(t) = V (t 0 ) α for all t 0 C. Therefore, f (V )(t) α for all t f (C) [actually for all t f (C) \ {y}]. So y s not a α semlmt pont of f (C). Hence f (C) s α semclosed set n Y. We now recall from [4] the defnton of α compact (crsp) set. Defnton 4.9. A crsp subset A of a fts. X s sad to be α compact f each α shadng of A by fuzzy open sets of X has fnte α subshadng. We now propose to defne α semcompact subset of a fts. Defnton A crsp subset A of a fts. X s sad to be α semcompact f each α shadng of A by fuzzy semopen sets of X has fnte α-subshadng.

5 Page 5 of 6 Theorem 4.3. A α semclosed subset A of a α semcompact space X s α semcompact. Proof. Let (X,τ) be a α semcompact space and A be a α semclosed subset of X. If x X \ A, then x s not a α semlmt pont of A. So there exsts a fuzzy semopen set U x n X such that U x (x) > α but U x (y) α for all y A [actually for all y A \ {x}]. Let Ω = {U x : x X \ A}. Let S be a α shadng of A by fuzzy semopen sets of X. Let W = Ω S. Clearly W s a α shadng of X by fuzzy semopen sets of X. As X s α semcompact, there exsts a fnte subset {W 1,W 2,W 3,...,W n } of W whch s agan a α shadng of X. Now the subset {W 1,W 2,W 3,...,W n } of W may contan some W whch are members of Ω. If we omt them we get a fnte α subshadng of S by fuzzy semopen sets of X consstng of the remanng members of W. Hence A s α semcompact. Corollary 4.1. Every α semclosed subset A of a α compact space X s α semcompact. Proof. As X s α semcompact, t s α compact. Hence by theorem (4.3), the corollary follows. Defnton A functon f : (X,τ) (Y,τ 1 ) s sad to be α semcontnuous f f 1 (A) s α semclosed set n X for every α closed set A n Y. Remark 4.3. In vew of theorem (4.1) every fuzzy semcontnuous functon s α semcontnuous functon. Theorem 4.4. Let f : (X,τ) (Y,τ 1 ) be a α semcontnuous functon. If A X s α semcompact set n (X,τ) then f (A) s α compact n (Y,τ 1 ). Proof. Let A be a α semcompact set n (X,τ). Let S be a α shadng of f (A) by fuzzy open sets n Y. For x A X, f (x) f (A) and there exsts U f (x) S such that U f (x) ( f (x)) > α. Let z / U 1 f (x) [0,α] then U f (x) S wth U f (x) (z) > α and U f (x) (y) α for all y U 1 [0,α]. Thus z s not a α lmt pont of U 1 [0,α] and hence U 1 [0,α] s α closed n Y. As f : X Y s f (x) f (x) f (x) α semcontnuous, f 1 (U 1 f (x) [0,α]) s α semclosed n X. Clearly x / f 1 (U 1 f (x) [0,α]). Indeed f x f 1 (U 1 [0,α]) then f (x) f (x) U 1 f (x) [0,α] whch mples U f (x)( f (x)) α whch contradcts the fact that U f (x) S [S s α shadng of f (A)]. Let C = U 1 f (x) [0,α], a α closed set n Y. So f 1 (C) s α semclosed set n X and as x / f 1 (C), x s not a α semlmt pont of f 1 (C). So there exsts fuzzy semopen set V x n X such that V x (x) > α and V x (z) α for all z f 1 (C). Now W = {V x : x A}s a α shadng of A by fuzzy semopen sets n X. As A s α semcompact, W has a fnte subset {V x1,v x2,v x3,...,v xn } whch s also a α shadng of A. That s, for every p A, V x (p) > α for some = 1,2,...,n. We clam that {U f (x ) : = 1,2,...,n} s a fnte α subshadng for f (A). Let s f (A), so f(t) = s for some t A. So V x (t) > α for some = 1,2,...,n. As t / f 1 (C), that s, as t / f 1 (U 1 [0,α]), we have f (t) / U 1 f (x ) f (x ) [0,α], whch mples that U f (x )( f (t)) / [0,α]. That s, U f (x )(s) / [0,α]. That s, U f (x )(s) > α. Hence {U f (x ) : = 1,2,...,n} s a fnte α subshadng for f (A) by fuzzy open sets n Y. Hence f (A) s α compact. Corollary 4.2. If f : (X,τ) (Y,τ 1 ) be a fuzzy semcontnuous functon and f A X s α semcompact n X then f(a) s α compact. Proof. Every fuzzy semcontnuous functon s α semcontnuous. So by theorem (4.4) the corollary follows. Theorem 4.5. Let (X,τ) be a fuzzy topologcal space. Then X s α semcompact ff every collecton R of α semclosed sets n X satsfyng the fnte ntersecton property has non-null ntersecton. Proof. Let (X, τ) be α semcompact. If possble, let there exsts a collecton R of α semclosed sets havng fnte ntersecton property but C = ϕ. Then (X \C) = X. Let x X. Then x X \C x for some C x R. Snce C x s α semclosed and x / C x, C R C R we have some fuzzy semopen sets U x n X such that U x (x) > α and U x (z) α for all z C x. Then S = {U x : x X} s an α shadng of X by fuzzy semopen sets n X. As (X,τ) s α semcompact, we have a fnte subcollecton {U x1,u x2,u x3,...,u xn } of S such that for every z X, we have U x (z) > α for some (1 n).thus X = n Ux 1 (α,1] and hence n {X \Ux 1 (α,1]} = ϕ. Let z C x. Then U x (z) α and hence z / Ux 1 Hence n (α,1]. Ths mples z {X \U 1 x (α,1]}. So C x {X \Ux 1 (α,1]}. C x = ϕ goes aganst the fnte ntersecton propety of R. Conversely, let S be an α shadng of X by fuzzy semopen sets n X. Let U α = {x X : U(x) α,u S}. If y / U α then U(y) > α. So we get a fuzzy semopen set U such that U(y) > α and U(z) α for every z U α. So y s not a α semlmt pont of U α. Hence U α s α semclosed set for each U S. Let R = {U α : U S}. Then R s a collecton of α semclosed sets. As S s a α shadng of X by fuzzy semopen sets of X, we have for each x X, there exsts V S such that V (x) > α. Hence x / V α for some V S. So U α = ϕ. Now by hypothess R does not satsfy the fnte ntersecton property. So there exsts a fnte subcollecton S 0 of U S S such that U α = ϕ. So for every x X there exsts U S 0 such that x / U α and hence U(x) > α. Thus X s α semcompact. U S 0

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