APPLICATIONS OF SEMIGENERALIZED -CLOSED SETS

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1 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM, N.RAJESH 2 Depatment of Mathematcs, Jeppaa Engneeng College, Chenna60099, gsm.maths@gmal.com 2 Depatment of Mathematcs, Raah Sefo Govt. College, Thanavu Pn 63005, Tamlnadu, Inda, naesh_topology@yahoo.co.n Abstact. In ths pape, we have study the some new popetes of sg closed sets n topologcal spaces. Keywods: Topologcal spaces, sg open sets. INTRODUCTION Genealzed open sets play a vey mpotant ole n Geneal Topology they ae now the eseach topcs of many topologsts woldwde. Indeed a sgnfcant theme n Geneal Topology Real analyss concens the vaous modfed foms of contnuty, sepaaton axoms etc. by utlzng genealzed open sets. Dmenson theoy plays an mpotant ole n the applcatons of Geneal Topology to Real Analyss Functonal Analyss. Recently, as a genealzaton of closed sets, the noton of sgclosed sets wee ntoduced studed by Raesh Ksteska [7]. In ths pape, we have study the some new popetes of sgclosed sets n topologcal spaces. 2. PRELIMINARIES Defnton 2. A subset A of a space ( X, ) s called semopen [3](esp. open [4]) f A Int(A)) (esp. A Int(Int(A)))). The complement of a sem open (esp. open) set s called semclosed (esp. closed). The semclosue [] of a subset A of X, denoted by sa), s defned to be the ntesecton of all semclosed sets contanng A n X. The closue of a subset s smlaly defned. Defnton 2.2. A subset A of a space X s called semgenealzed closed (befly closed) [7] f A) U wheneve A U U s semopen n X. The complement of closed set s called sgopen. sg sg

2 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) The unon (esp. ntesecton) of all sgopen (esp. sg closed) sets each contaned n (esp. contanng) a set A n a space X s called the sg nteo ( esp. sg closue) of A s denoted by sg Int(A) (esp. sg A)) [7]. The famly of all sg open (esp. sg closed) sets of, sg closed) sets of X, X s denoted by sg O(X) (esp. sg C(X)). The famly of all sg open (esp. contanng a pont x X s denoted by sg O(X, x) (esp. sg C(X, x)). It s well known that sg O(X) foms a topology [7]. Defnton 2.3. [2] A famly { A : } of subsets of a space X s sad to be locally fnte famly f fo each pont x of X, thee exsts a neghbohood G of x such that the set { : G A } s fnte. Lemma 2.4. [2] If { A : } s a locally fnte famly of subsets of a space X, then the famly { A ) : } s a locally fnte famly of X A )= A ). Defnton 2.5. [2] A famly { A : } of subsets of a space X s sad to be pontfnte f fo each pont x of X, the set : x A } s fnte. { Defnton 2.6. [2] An open cove { G : } of a space X s sad to be shnkable f thee exsts an open cove { : } H of X such that H G Cl fo each. Defnton 2.7. [5] The famly { A : } { B : } of subsets of a set X ae sad to be smla, f fo each fnte subset of, the sets A B ae ethe both empty o both nonempty. Theoem 2.8. [2] Let X be any topologcal space. The followng statements ae equvalents: () X s a nomal space. () Each pontfnte open cove of X s shnkable. () Each fnte open cove of X has a locally fnte closed efnement. Theoem 2.9. [5] Let { U : } be a locally fnte famly of open sets of a nomal space X { F : } a famly of closed sets such that F U fo each. Then thee exsts a famly { G : } of open sets such that F G Cl G U fo each the famles { F : } { Cl : } ae smla. G Poposton 2.0. [7] Fo subset A A (I) of a space X,, the followng hold: () A sg ClA () If A B, then sg ClA sg ClB. () sg Cl{ A : I} { sg Cl A : I}. (v) sg Cl{ A : I} { sg Cl A : I} 2

3 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) Defnton 2.. If A s a subset of a space X, then the sg A)\ sg Int(A) s denoted by sg Bd(A). sg bounday of A s defned as 3. sg NORMAL SPACE Defnton 3.. A topologcal space X s sad to be sg nomal [6] f wheneve A B ae dsont sg closed sets n X, thee exst dsont sg open sets U V wth A U B V Defnton 3.2. A famly A : of subsets of a space X s sad to be sg locally fnte famly f fo each pont x of X, thee exsts an sg open set G of X such that the set : G A s fnte Defnton 3.3. An open cove G : of a space X s sad to be sg shnkable f thee exsts an sg open cove H : of X such that sg H ) G fo each. Lemma3.4. If A : s a locally fnte famly of subsets of a space X, then the famly { sg A ): } s sg locally fnte famly of X. Moeove, sg A ) sg A ). Poof. Fom Lemma 2.4, we obtan that the famly { A ): } s a locally fnte famly of X wheneve { A : } s locally fnte. Snce sg A ) A ) fo evey, the famly { sg A ): } s a locally fnte famly of X. To pove that sg A ) sg A ), we have sg A ) sg A ).Theefoe, t s suffcent to pove that sg A ) sg A ). Suppose that x sg A ), so x sg A ), fo all. Ths means that thee exsts an sg open set G such that G A fo all hence, G sg Cl ( A ) fo all. Snce the famly { A : } s locally fnte, thee exsts an open set H whch contans x the set : H A s fnte. Ths means that thee exsts a fnte subset M of such that H A fo M. Fom above we obtan that x belongs to X \ sg Cl ( A ) fo evey. Snce the famly of sg open sets foms a topology on X [7], the set V X \ sg Cl ( A ): s an sg open set n X contanng x. But H V s an sg open set contanng x ( H V ) fo all. Theefoe ) ( H V ) ( A. Ths mples that x sg A ) ths completes the poof. Theoem 3.5. Let X be any topologcal space. Then the followng statements ae equvalent: () X s an sg nomalspace. (2) Each pont fnte open cove of X s sg shnkable. (3) Each fnte sg open cove of X has a locally fnte sg closed efnement A 3

4 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) Poof. () (2): Let U : be a pont fnte open cove of an sg nomal space X, we may assume s well odeed. We shall constuct an sg shnkable famly of U : by tansfnte nducton. Let suppose that fo each we have sg open set V such that sg V ) U fo each v, we have ( V ) ( U ) X. Let x X, then snce U : s a pontfnte, thee exsts the v v lagest element such that x U. If, then x U f, then x V V. Hence, ( V ) ( U ) X. Thus, U contans the complement of ( V ) ( U ) that X [ ( V ) ( U )] \. Snce X s nomal sg, thee exsts an sg open set V such V sg Cl ( V ) U. Thus, sg V ) U ( V ) ( U ) X. Hence the constucton of an sg shnkable famly fo { U : } s completed by tansfnte nducton. (2) (3) : Obvous. (3) () : Let X be a space such that each fnte sg open cov e of X has a locally fnte sg closed efnement. Let A B be two sg closed sets n X. The sg open cov e X \ A, X \ B of X has a locally fnte sg closed efnement. Let E be the unon of membes of dsont fom A let F be unon of membes of dsont fom B. Then, by Lemma 3.4, E F ae sg closed sets E F X. Thus, f U X \ E V X \ F then U V ae dsont sg open sets such that A U B V Theefoe, X s an sg nomal space. Poposton 3.6. Let sg sg nomal space X F a famly of closed sets F U fo each. Then thee exsts a famly sg open F G sg Cl ( U ) U the famles F Cl ( G ) U be a locally fnte famly of open sets of an sg such that G of sets such that sg ae smla. Poof. Let be wellodeed wth a least element. By tansfnte nducton, we shall constuct a famly G of sg open sets such that F G sg Cl (G ) U fo each element n the famly G ) f K F f F. Suppose that that G ae defned fo such that fo s smla to each the famly K s smla F. Let L G F ) f f L be the famly gven by: 4

5 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) Then F, 2,...,... only f famly L. Fo suppose that,...,, then L F because 3.4, E : E F, 2 K. Theefoe K s smla to L f F. Snce L G fo each, the L s locally fnte. Thus f s the set of fnte subsets of fo each, E L, then E s locally fnte famly of sg closed sets. Hence by Lemma E s an sg closed set whch s dsont fom F. Theefoe, thee exsts an sg open set G such that F G sg Cl ( G ) U sg Cl ( G E). Now the sg open set G s defned fo to complete the poof t emans to show that the famly K s smla to the famlyf. It s suffcent to show that the famles L ae smla. Suppose that,..., that, 2 K L, we have to show that K Suppose that f thee s nothng to pove. If, then L... L F L... L. Hence by the constucton L... L sg Cl ( G L )... L. Thus K 4. sg COVERING DIMENSION In ths secton, we ntoduce a type of a coveng dmenson by usng we call the sg coveng dmenson functon. sg open sets whch Defnton 4.. The sg coveng dmenson of a topologcal space X s the least postve ntege n such that evey fnte sg open cove of X has an sg open efnement of ode not exceedng n o s f thee s no such ntege. We shall denote the sg coveng dmenson of a space X by dm. If X s an empty set, then dm X X sg dm X n f each fnte sg open cove of X has an sg open efnement of ode sg not exceedng n. Also we have dmsg X n f t s tue that dmsg X n but t s not tue f dmsg X n. Fnally, dmsg X f fo evey ntege n thee exsts a fnte sg open cove whch has no sg open efnement of ode not exceedng n. Poposton 4.2. If Y s an open a closed subset of a space X, then dm Y dm sg sg sg X 5

6 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) Poof. It s suffcent to pove that f dmsg X n, then dmsg Y n. Let U, U2,... U k be an sg open cove of the open set Y. Then U s sg open n X fo each snce evey open set s sg open. Then the fnte sg open cove U, U2,... Uk, X \ Y of X has an sg open efnement sg of ode whch not exceedng n.let be all membes of sg except those membes assocated wth X\Y, snce evey open set s sg open, then each membe of s sg open n Y also s a efnement of U, U2,... U k of ode not exceedng n. Ths mples that dm sg Y n. Now we gve some chaactezatons of the sg coveng dmenson n topologcal spaces. Theoem4.3. If X s a topologcal space, then the followng statements about X ae equvalent: () dm X n sg (2) Fo any fnte sg open cove U, U2,..., U k, V of X thee s an sg open cove V 2,..., V k of ode not exceedng n such that V U fo,2,..., k (3) If U U, 2,..., U n 2 s an sg open cove of X, thee s an sg open cove V, V 2,..., V n 2 such that n V 2 U V Poof. () (2) : Suppose that dm X n the sg open cove U, U2,..., U k of X sg has an open efnement of ode not exceedng n. If W, then W U fo some. Let each W n be assocated wth one of the sets U contanng t let V be the unon of those membes of thus assocated wth U, snce evey open set s sg open, then V s sg open V U each pont of X s n some membe of hence n some V. Each pont x of X s n at most n+ membes of sg, each of whch s assocated wth a unque V V s an sg open cove U hence s n at most n+ membes of. Thus of X of ode not exceedng n. (2) (3) : Obvous. (3) (2) : Let X be a space satsfyng (3) U, U2,..., U k a fnte sg open cove of X, we can assume that k > n+. Let G U f n, then G, G 2,..., G n 2 s an sg open cove of X k G n 2 U n2 so by hypothess thee s a n 2 sg open cove H, H2,..., H k such that H G H. Let W U f n let W U Gn2 f n. Then W W,..., s an sg open cove of X each W U n 2 W. If thee exsts, 2 W k such set B of {,2,,k} wth n+2 elements such that n 2 W, let the membes of be 6

7 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) enumbeed to gve a famly P P P,..., to P, we obtan the n 2 ', 2 P n k 2 P. By applyng the above constucton ' ' ' ' sg open cove W W, W,..., W such that W P W. Thus by a fnte numbe of epettons of ths pocess we obtan an sg open cove V V,...,, 2 V k of X, of ode not exceedng n such that V U, (2) () : Obvous. Poposton 4.4. In a topologcal space X f dmsg X 0, then X s a sg nomal Space. Poof. Suppose that dmsg X 0 let F E be any two dsont sg closed sets n X, then X \ F,X \ E s an sg open cove of X ; hence, thee exsts a sg open efnement of ode not exceedng 0. Ths means that all membes of ae pa wse dsont. Let G be the unon of all membe of that ae assocated wth X\E H be the unon of all membes of that ae assocated wth X\F, hence, G H ae sg open sets such that G H X,G X \ E H X \ F G H. Thus G H ae dsont sg open sets such that F G E H. Hence X s a sg nomal space. In sg nomal spaces, sg cov eng dmenson can be defned n tems of the ode of fnte sg closed efnements of fnte sg open cove. Poposton4.5. If X, s a topologcal space, then the followng statements about X ae equvalent. () dm X n. sg (2) Fo any fnte sg open cov e U, U2,..., U k sg open cov e V, V2,..., V k such that sg Cl (V ) U sg Cl (V ), sg Cl (V2 ),..., sg Cl (Vk ) does not exceed n sg open cov U, U2,..., U k F F,..., such that F U the ode fo F F,..., (3) Fo any fnte e, 2 F k ' 2 k of X thee s a the ode fo of X thee s a sg closed cove does not exceed n. (4) Evey fnte sg open cov e of X has a fnte sg closed efnement of ode not exceedng n. U, U2,..., U k s an sg open cov e of X, thee s an sg closed cove (5) If n F, F2,..., F k such that F 2 U F. Poof. () (2): Suppose that dm sg X n, let U, U2,..., U k be sg open cov e of X, then by Theoem 4.3, thee exsts an sg open cov e W, W2,..., W k of ode not exceedng n such that W U. Snce X s sg nomal, by Theoem 3.5, thee exsts an sg open cov e V, V2,..., V k such that sg Cl (V ) W fo each. Then V, V2,..., V k, 2 F k 7

8 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) s an sg open cov e wth the equed popetes. (2) (3) (3) (4): Obvous. (4) (5): Let X be a space satsfyng (4) let U, U 2,..., U n 2 be sg open cov e of X. Then the cove has a fnte sg closed efnement v of ode not exceedng n. If the sets U, then n 2 E v, then U contanng t let F s sg closed, F U E U fo some. Let each E n v be assocated wth one of F be the unon of those membes of v whch assocated wth F s a sg cov e of X such that, F2,..., F n 2 F.(5) () : Let X be a space satsfyng (5) let U, U 2,..., U n 2 be an sg open cov e of X, by hypothess thee exsts an sg closed n cove F F, 2 F n 2,..., such that each F U 2 F. By Poposton 3.6, thee exst sg open sets V, V 2,..., V n 2 such that F V U fo each V s smla to F. Thus V, V 2,..., V n 2 s an sg open cov e of X, eachv 2 U by Theoem 4.3, dm X n sg n V. Theefoe Poposton 4.6. If X s an sg nomal space, then the followng statements about X ae equvalent: () dm X n, sg (2) Fo any fnte sg closed sets F, F... } each famly of sg open sets { 2, U n { 2, F n U, U... } such that F U thee exsts a famly V, V... } of sg open sets such that F V sg ) (3) Fo each famly sg closed sets } { 2, V n n V U fo each, BdV { F, F2,... Fk each famly of sg U { V, V2,... Vk open sets { U, U2,... Uk} such that F, thee exsts a famly } { W, W2,... Wk } of sg open sets such that F V sg V ) W U fo each the ode of the famly { sg W ) \ V, sg W2 ) \ V2... sg W k ) \ Vk } does not exceed n. (4) Fo each famly sg closed sets F, F... F } each famly of sg open sets {, U2,... Uk} { 2, k U such that F U, thee exsts a famly V, V... V } of sg open { 2, k sets F V sg V ) U the ode of the famly { sg Bd( V ), sg Bd V ),..., sgbd( V ) does not exceed n. ( 2 k Poof. () (2): Suppose that dm X n, let F, F... } be sg closed sets sg { 2, F n { U, U2,... U n } sg open sets such that F U. Snce dm X n, the sg open n cove of X consstng of sets of the fom 2 H, whee H U o H X \ F fo each, has a fnte sg open efnement W, W... W } of ode not exceedng n, Snce X s sg { 2, q sg 8

9 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) nomal, thee s an sg closed cove K, K... K } such that K W fo,2,,,,,,, q. Let { 2, q N denoted the set such that F W. Fo,2,,,,,,, q, we can fnd sg open V fo n sets N such that K V sg V ) W sg V ) W f.now fo each,2,.., n, letv { V : N}. Then V s sg open F V fo f x F x K ; then N so that x V V. Futhe moe f N so that F W then W s not contaned n X \ F so that W U. Thus, f N, then V snce sg V ) { sg V ) : N }, t follows that sg V ) U. U n 2 Fnally suppose that x sg Bd( V ) snce sg Bd V ) { sgbd( V ) : N }, t ( follows that fo each thee exsts such that xsg Bd( V ) f, then fo f, then x sg V ) x sg V ) but x V x V whch ae absud, snce ethe sg V ) V o sg V ) V. Fo each, x V so that x But { K } s a sg cove of X so thee exsts 0 dffeent fom each of the K. such that x K W. Snce x V, t follows that n V n 0 o o x fo, 2,..., n so that x. Snce the ode of { W } does not exceed n, ths s absud. Hence sg Bd( V ). (2) (3): Let (2) hold, let F, F2,..., Fk be sg closed sets let U, U2,... U K be sg open sets such that F U. We can assume that k n ; othewse, thee s nothng to pove. Let the subset {,2,..., k } contanng n elements be enumeated as C, C2,..., C q, whee q kc n. By usng (2), we can fnd sgopen sets V, fo n C such that n, sg V, U F V ) sg Bd( V, ) W n. We have a fnte famly { sg Bd( V, ) : C} of sg closed sets of the sg nomal space X sg Bd( V, ) U fo each nc. Thus, by Poposton 3.6, fo each nc, thee exsts an sg open set G such that sg Bd( V, ) G sg G ) U sg G ) C s smla to sg ( ), so that n patcula sg ). Let W, V, G f C, then Bd V, C C sg V W sg Cl ( W. ) U snce sg C l( W ) \ V ) sg C l( G ), we have C, ), G (,, ( sg W, \ V,). If C, let V, be an sg open set such that F V, sg V, ) U let W, U. Then fo, 2,..., k we have sgopen sets V, W, such that F V, sg V, ) U ( sg W, \ V, ). Suppose that m q fo,2,..., k we fnd sg open sets V, m W, m such that V sg V W U sg W ) \ V ) f m. F, m, m), m By the above agument we can fnd c (, m, m c sg open sets V, m W, m such that sg Cl ( V ),m 9

10 Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : Volume Issue 4 (Apl 202) V, m V, m) W, m W, m sg sg W ) \ V ) Snce sg W, m) \ V, m (, m, m cm ( W, m) \ V, m c sg W ) \ V ). We have sg ) f m. (, m, m Thus by nducton fo,2,..., k, we can fnd sg open sets V W V W espectvely). Such that F V sg V ) W U (, q, q ( sg W ) \ V ), fo,2,..., k. Thus the ode of the famly { sg Bd( W \ V ),..., c sgbd W k \ V )} does not exceed n. (3) (4): Obvous. (4) (): Let (4) hold let ( k { U, U2,... U n 2} be an closed cove {, F2,... F n } famly of {, V2,... V n the famly { sg Bd( V ), sgbd( V2 ),... L sg open cove of X. Snce X s sg nomal, thee exsts an sg F of X such that F U fo each. By hypothess thee exsts a sg open sets V } such that F V sg V ) U fo each,.., sgbd( V n 2) has ode not exceedng n. Let sg V ) \ V fo,2,..., n 2. Fo each, L s an sg closed, L, L... } s an sg closed cove of X, fo f x X, thee exsts such that { 2, L n 2 x V xv fo so that x L. Now L sg V ) ( X \ V ) so that n2 L n2 n n sg V ) ( X \ V ) sgbd( V ). Thus L, L,,,,,,, }, s an sg closed cove of X, dm X n. sg n V U 2 L sg ) { 2 L n 2 L = Hence by poposton 4.5, REFERENCES. Cossley SG Hldeb SK. Semclosue. Texas J. Sc., 97; 22: J. Dugund, Topology, Allyn Bacon Inc. Boston Levne N. Semopen sets semcontnuty n topologcal spaces. Ame. Math. Monthly., 963;70: Nastad O. On some classes of nealy open sets. Pacfc J. Math., 965;5: A. R. Peas, Dmenson Theoy of Geneal Spaces, Cambdge, Unvesty pess, Cambdge N.Raesh G. Shanmugam. On sgegula space sg nomal space (unde pepaaton). 7. Ra esh N Ksteska B. Semgenealzed closed sets, Antactca J. Math., 6() 2009, 2. 0