E be a set of parameters. A pair FE, is called a soft. A and GB, over X is the soft set HC,, and GB, over X is the soft set HC,, where.

Transcription

1 The Exteso of Sgular Homology o the Category of Soft Topologcal Spaces Sad Bayramov Leoard Mdzarshvl Cgdem Guduz (Aras) Departmet of Mathematcs Kafkas Uversty Kars 3600-Turkey Departmet of Mathematcs Georga Techcal Uversty Tbls Georga Departmet of Mathematcs Kocael Uversty Kocael 4380-Turkey Abstract: ths artcle we troduce sgular cubc homology groups of soft topologcal spaces beg exteso of the relevat homology groups of topologcal spaces For ths purpose the cocept of soft ut terval s tally troduced Further usg the cocept of soft ut terval defto the homotopy relato the category of soft topologcal spaces s gve ad the soft homotopc varace of the etered homology groups s proved NTRODUCTON May practcal problems ecoomcs egeerg evromet socal scece medcal scece etc caot be dealt wth by classcal methods because classcal method have heret dffcultes The reaso for these dffcultes may be due to the adequacy of the theores of parameterzato tools Molodtsov [7] tated the cocept of soft set theory as a ew mathematcal tool for dealg wth ucertates Research works soft set theory ad ts applcatos varous felds have bee progressg rapdly sce Maj et al ([4] [5]) troduced several operatos o soft sets ad appled t to decso makg problems The dea of soft topologcal spaces was frst gve by M Shabr M Naz [8] ad mappgs betwee soft sets were descrbed by P Majumdar SK Samata [6] Later may researches about soft topologcal spaces were studed [ ] these studes the cocept of soft pot s expressed by dfferet approaches the study we use the cocept of soft pot whch was gve [9] Soft topologcal spaces ad soft cotuous mappgs form category ad ths category s exteso of category of topologcal spaces Also category of fuzzy topologcal spaces s exteso of category of topologcal spaces t s kow that research of topologcal spaces the mportat place s take by methods of the algebrac topology [56] Ufortuately these methods were ot wdely used the research of fuzzy topologcal spaces although there were some artcles [789202] our opo such stuato was due to defto of fuzzy as of a ut terval category of soft topologcal spaces methods of algebrac topology were't cosdered yet ths artcle we troduce sgular cubc homology groups of soft topologcal spaces beg exteso of the relevat homology groups of topologcal spaces For ths purpose the cocept of soft ut terval s tally troduced Further usg the cocept of soft ut terval defto the homotopy relato the category of soft topologcal spaces s gve ad the soft homotopc varace of the troduced homology groups s proved PRELMNARES Throughout ths paper X refers to a tal uverse E s the set of all parameters for X We ow recall some deftos Defto 2 ([7]) Let X be a tal uverse set ad E be a set of parameters A par FE s called a soft set over X f ad oly f F s a mappg from E to the set of all subsets of the set X e F : E P X P X s the power set of X where Defto 22 ([5]) The tersecto of two soft sets F A ad GB over X s the soft set HC where C A B ad C H F G Ths s deoted by F A G B H C Defto 23 ([5]) The uo of two soft sets F A ad GB over X s the soft set HC where C A B ad C F f A B H G f B A F G A B F A G B H C Ths s deoted by F A ad GB deoted by F A G B f Defto 24 ([5]) Let soft sets over X The () A B F a G a for each a A (2) GB be two F A s called a soft subset of 292

2 Defto 25 ([5]) A soft set to be a absolute soft set deoted by F X Defto 26 ([5]) A soft set FE over X s sad X f for all E FE over X s sad to be a ull soft set deoted by f for all F Ø E Defto 27 ([8]) Let be a collecto of soft sets over X The s sad to be a soft topology o X f () X belog to (2) the uo of ay umber of soft sets belogs to (3) the tersecto of ay two soft sets belogs to The trplet over X s called a soft topologcal space Defto 28 ([9]) Let FE be a soft set over X ad x X The soft set FE s called a soft pot deoted by x e E f for the elemet e E F e x ad Fe Ø for all ee e ths case we say that x e E s a pot of a soft set FE Defto 29 ([2]) Let E ad Y E be two soft topologcal spaces f : Y E be a mappg For each soft eghbourhood H E of exsts a soft eghbourhood that f x E f there e FE of x E such f F E H E the f s sad to be soft cotuous mappg at x e E Defto 20 ([0]) Let XT be a famly of soft topologcal spaces The the tal soft topology o X called product soft topology o X (Here X geerated by the famly p q soft projecto mappg from X to e q X ) s p s the THE MAN RESULTS Soft topologcal spaces ad ther soft cotuous maps form a category Ths category s deoted by Stop Each usual topologcal spaces are obtaed soft topologcal spaces Actually let ( X ) be ay topologcal spaces We cosder of the sgle-pot parameter set E {} For every U the fucto F : E P( X ) U s defed as F () U The famly { } U F U U forms a soft topology over X Thus every topologcal space ca be hadled a soft topologcal space So the category of topologcal spaces s a subcategory the category of soft topologcal spaces Uless otherwse stated E N wll be assumed to be a set of parameters ad the set of ratoal umbers o the closed terval 0 wll be cosdered as the set r f for all N ad for all 0 we defe the soft set F : N P as F r r 0 The the famly F N s a soft base of soft topology o ad s called a soft topology geerated by Defto 3 A soft topologcal space called a soft ut terval Defto 32 Let N s be a soft topologcal space The a soft sgular cube soft cotuous map T : N X E where T : X : N E Here topologcal product of soft topologcal space ad deotes the soft product topology [0] S X E cubes s a s the N deotes the set of all soft sgular Defto 33 A soft sgular cube T S X E ( ) s called degeerate f oly f the mappg the coordate values Defto 34 Let T does ot deped o oe of be a soft topologcal space ad be the (usual) addtve group of tegers 293

3 Q X E For group geerated by the set S X E sgular cubes X E For let D X E of Q X E deotes the free abela (b) of all soft deote the subgroup geerated by all degeerate soft sgular cubes ad D X E 0 0 The we ca defe Q X E C X E 0 D X E ad called t the group of sgular chas X E For all ad k N soft mappgs we defe the followg 0 k k k k A A : N N by the formula: k j j j j k j j 0k A x j x 0 j x j x j k x j x j k A x j x j x j x j k x j x j t s clear that the soft mappgs 0 A k k k k A are soft cotuous [2] For Defto 35 Let T S X E 2 0 k k 0 k k A T T A ad A T T A S E k k k k are called the th lower face ad th upper face of T respectvely Proposto 36 Let T S X E j The (a) ad A A T A A T 0 k k 0 l l 0 l l 0 k k s s s s A A T A A T k k l l l l k k s s s s (c) 0 k k l l l l 0 k k A As s T As s A T (d) k k 0 l l 0 l l k k A As s T As s A T Proof (a) For ay T S X E j j2 j j2 0 k k 0 l l 0 k k 0 l l A As s T x x A T As s x x we have 0k k T A x j x 0 js l x j x s j2 T x j x 0 0 j k x j x js l x j x s j2 j j2 j j2 0 l l 0 k k 0 l l 0 k k As s A T x x As s T A x x 0l l T As s x j x 0 j k x j x j2 T x j x 0 0 j k x j x js l x js Hece (a) holds Smlarly (b) (c) (d) are easly checked Defto 37 Let spaces We defe a homomorphsm be a soft topologcal Q X E Q X E such that for T S X E : x j 2 k k 0 k k T A T A T Lemma 38 Let we have 0 Proof We show oly that for ay T S X E T 0k deed sce 294

4 0 k k k k k k T k k A T A s T s j j j s s k k 0 k k A T A T s k k k k 0k k k k A A A A k k 0k k 0k k 0k k As s A As s A T k k k k 0k Thus k k k A A A A s k k 0k k 0k k 0k k s s s s A T x x x A T s s s s s k k k k s s j j j s s s s s s s s s ad k k 0k k As s T As s T Hece T D s obtaed A A A A T k k k k k k k k * : C A As s T As s A T C for s where ad may be ether 0 or X E the f we deote s s j k k k k A A T s s s the s s s s s obtaed s s s k k k k A As s T s k k k k A As s T * Now let f g: Y E k k k k k k k k A A T A A T s s s s T Hece s obtaed 0k Proposto 39 Let ( D ) D Proof Let T D X E the The we shall prove oly that for T D X E T D X E deed suppose that ot deped o y j j T x x x j j j x The for s does x j A T x x x A T s satsfed For s ad x j y As a cosequece of Proposto 39 homomorphsm j Thus for the soft topologcal space duces a followg cha complexes of groups * : C X E C X E C X E be a soft cotuous mappg of soft topologcal spaces We defe a homomorphsm f g : S SY E such that for ay T S X E f g T f g T f T g S Y E * The followg proposto ca be easly checked Proposto 30 For 0 f g D X E D Y E * From the Proposto 30 f g * duces a homomorphsm f g C X E C Y E : * Proposto 3 Let space The the correspodece E C X E be a soft topologcal f g f g C X E C Y E : * 295

5 s a fuctor from the category of soft topologcal spaces to the category of cha complexes Proof Now we oly show that f g * f g * * f g T * f g T * k 0 k f g T A f g T A k 0k f g * T A T A k 0 k * f g * T * f g A T A T Thus f g f g s satsfed * * Aother codtos ca be easly obtaed Defto 32 Let space We defe as Z X E ker be a soft topologcal * B X E m * ad call them the group of sgular cycles ad sgular boudares X E respectvely The group Z X E H X E B X E s called the sgular homology group of soft topologcal space X E Theorem 33 Let E {} ad ( X ) be the usual topologcal space The H [( ) Z] H ( X Z) 0 where H( X Z ) s the dmesoal cubcal sgular homology group of X (see [6]) t s clear that for each soft mappg f g : Y E f g Z X E Z Y E * ad f g B X E B Y E * The the homomorphsm * homomorphsm f g duces a f g H X E H Y E : ** Theorem 34 The correspodece X E H X E ** f g f g H X E H Y E : s a fuctor from the category of soft topologcal spaces to the category of groups Now we show that homology fuctor s homotopc varat the category of soft topologcal spaces For ths we frstly gve soft homotopy relato the category of soft topologcal spaces Defto 35 Let E Y E be two soft topologcal spaces ad f g : Y E be two soft cotuous mappg f there exsts a soft cotuous mappg F : N Y E such that F x 0 f x F x g x the we say that e e e e f s soft homotopc to Hece we have the followg example g Example 36 Let g : N N be two soft N 5 2 F xe t t xe t 2 the mappgs ad defe g t f we take N s soft homotopc to g the followg we gve some propertes of soft homotopy relato Theorem 37 The soft homotopy relato the category of soft topologcal spaces s a equvalece relato Proof t s obvous that ths relato s reflexve ad symmetrc We show that soft homotopy relato s trastve 5 296

6 Let f be soft homotopc to g ad g f g : X e Y h homotopc for each e E [2] f s soft homotopc to g the there f g : Y E F such that be soft homotopc to Sce exsts a soft cotuous mappg homotopc F xe0 f xe F xe g xe F : N Y E Sce g s soft homotopc to h the there homotopy betwee these mappgs For each e E exsts a soft cotuous mappg G such that N F : X e Y e G xe0 g xe G xe h xe satsfed: f we take soft cotuous mappg H such that F x F x f x f x H x t the e F xe 2 t 0 t 2 G xe 2t t 2 f s soft homotopc to h Theorem 38 The composto operato the category of soft topologcal spaces s varat wth respect to the soft homotopy relato Let E Y E be two soft topologcal spaces Cosder the product soft topologcal spaces as X Y E E [0] A base of soft topology cossts of the soft sets F E G E F E G E The for each F Ge e F e G e e e E E s a soft ope set X Y The followg proposto easly obta Proposto 39 Let E Y E be two soft topologcal spaces The X Y X Y ee e e holds Proposto 320 Let E Y E be two soft topologcal spaces ad f g : Y E be two soft mappgs f g the f s soft homotopc to are the mappgs e Proof Let the mappgs be soft ad be a ad s a cotuous mappg ad the followg codtos are 0 0 e e F x F x g x g x e e Theorem 32 Let f g : Y E be two soft mappgs f g the f s soft homotopc to the morphsms of cha complexes f g : C X E Z C Y E * * s cha homotopc [56] Proof F : N Y E Let be a soft homotopy betwee the soft mappgs f ad g We wll use F to costruct a sequece of homomorphsms D : C X E Z C Y E Z 0 such that D D f g * * For 0 we defe C X E Z 0 C Y E Z 0 D : C X E Z C Y E Z are the zero homeomorphsms Frst of all we defe a sequece of homeomorphsms D : C X E Z C Y E Z ad 0 as follows For ay T S X E D T S Y E by the formula: defe D T x x F T x x x 297

7 We wat to compute DT A D T f T A 0 D T g T * * A D T D A T A D T D A T Now we have 0 D T A D T A D T 2 0 f T g T D A T A T * * 2 j 0 f T g T D A T A T * * j j j f T g T D T * * Therefore for ay T S X E D T D T g T f T * * () Now we see that f cube the D cube Hece T s a degeerate soft sgular D D X E D Y E ad therefore T s a degeerate soft D duces a homomorphsm D : C X E Z C Y E Z t follows from () that the famly of homomorphsms D s a cha homotopy betwee * g * f ad Corollary 322 Let f g : Y E be a soft homotopc mappgs The f g : H ** ** X E Z H Y E Z for all 0 V CONCLUSON ths artcle we troduce sgular cubc homology groups of soft topologcal spaces beg exteso of the relevat homology groups of topologcal spaces For ths purpose the cocept of soft ut terval s tally troduced Further usg the cocept of soft ut terval defto the homotopy relato the category Observe that of soft topologcal spaces s gve ad the soft homotopc varace of the etered homology groups s proved As follow up of ths paper we would lke to work o the sgular homology theory o the category of soft topologcal spaces also to prove axom of the cotuty for ths theory We beleve that ths work wll be a ctemet for applcato of methods of algebrac topology soft topologcal spaces research ad other sceces REFERENCES [] M Al F Feg X Lu WK M ad M Shabr o some ew operatos soft set theory Comput Math 298 Appl49 (2005) [2] C Guduz (Aras) A Somez H Çakallı O Soft Mappgs arxv: v [mathgm] 6 May 203 [3] S Hussa ad B Ahmad Some propertes of soft topologcal spaces Computers ad Math wth Applcatos 62(20) [4] P K Maj ARRoy R Bswas A applcato of soft sets a decso makg problem Comput Math Appl44 (2002) [5] PKMaj RBswas ARRoy Soft set theory Comput Math Appl45 (2003) [6] P Majumdar ad S K Samata o soft mappgs Comput Math Appl 60 (200) [7] D Molodtsov Soft set-theory-frst results Comput Math Appl37(999) 9-3 [8] M Shabr ad M Naz O soft topologcal spaces Comput Math Appl6 (20) [9] S Bayramov C Guduz(Aras) Soft locally compact ad soft Para compact spaces Joural of Mathematcs ad System Scece (accepted) [0] AAyguoglu HAygu Some otes o soft topologcal spaces Neural Computg ad Applcatos 2- (202) 3-9 [] Zorlutua M Akdag WK M ad S Atmaca Remarks o soft topologcal spaces Aals of Fuzzy Mathematcs ad formatcs 3(2) (202) 7-85 [2] BP Varol H Aygu O soft Hausorff spaces Aals of Fuzzy Mathematcs ad formatcs5() (203) 5-24 [3] N Cagma S Karatas S Egoglu Soft topology Comput Math Appl 62 (20) [4] WK M A ote o soft topologcal spaces Comput Math Appl 62 (20) [5] S Eleberg N Steerod Foudatos of Algebrac Topology Prceto Uversty Press Prceto New Jersey 952 [6] WS Massey Sgular Homology Theory Sprger New York 980 [7] Lu Wag-j Zheg Chog-you Sgular homology groups of fuzzy topologcal spaces Fuzzy Sets ad Systems 5 (985) [8] AR Salleh The homotopy property of the duced homomorphsms o homology groups of fuzzy topologcal spaces Fuzzy Sets ad Systems 56 (993) -6

8 [9] C Guduz(Aras) S Bayramov O fuzzy exact homotopy sets Southeast Asa Bulleto of Mathematcs 34 (200) [20] C Guduz(Aras) S Bayramov Cech homology theory of Sostak fuzzy topologcal spaces teratoal Joural of Cotemporary Math Sceces No 2 (200) [2] S Bayramov C Guduz(Aras) TY Ozturk Homology theory the category of fuzzy topologcal spaces Lambert Academc Publshg