
filter (kanakk)

ganithashaasthrathil, "valia" allengil "aathyanthika" gadakangale vivarikkunna bhaagikamaayi krameekaricha settinte (poosatt) oru prathyeka upavibhagamaanu filter allengil oordar filter. filterukal kramatthilum laattis sidhaanthathilum prathyakshappedunnu, mathramalla topology, avaa evide ninnaan uthbhavikkunnath. filterilekku eratta enna ashayam oru oordar idealaanu. 
filterukalude prathyeka kesukalil ultrafilterukal ulppedunnu, avaa valuthakkaan kazhiyaatha filterukalaanu, koodaathe ganithashaasthra yukthiyil nirmmaanaparamallaatha saankethikavidyakale vivarikkunnu. 
1937ll henry carton settukalile filterukal avatharippichu. nicolas borbacki avarude topology general enna pusthakatthil iii ennathinu pakaramaayi filterukale janapriyamakki. echu. moorum herman ell. 1922-ll smithinte nettu enna ashayam; oordar filterukal eee ashayathe ulppeduthiyittulla oru power settinte nirddishta kesil ninnu ekapaksheeyamaayi bhaagikamaayi oordar cheytha settukalilekku saamaanyavalkkarikkunnu. ennirunnaalum, power-settu filterukalude sidhaantham athintethaaya thaalparyam nilanirthunnu, bhaagikamaayi topologyil ganyamaaya prayogangalkkaayi. 
prachodanam.
bhaagikamaayi oordar cheytha oru settu (poosatt) shariyaakkuka. avabodhapoorvvam, chila maanadandangal niravettaan paryapthamaaya gadakangalulla amgangalude oru upavibhagamaanu filter. udaaharanathinu, mukalilulla gadakangalude koottam oru filteraanengil, athine principal filter ennu vilikkunnu. (iva thaarathamyappeduthaanaavatha gadakangalaanenkil, pradhana filter mattonnilum adangiyittilla.)
athupole, oru settile oru filteril nalkiyirikkunna chilath ulkkollaan paryapthamaaya valuppamulla upavibhagangal adangiyirikkunnu. udaaharanathinu, yadhaartha line aanenkil, avayude interior ulppedeyulla settukalude kudumbam oru filteraanu, athine ayal filter ennu vilikkunnu. eee saahacharyathil ithu athinekkaal alpam valuthaanu, pakshe athil ippozhum variyude mattoru prathyeka pooyintum adangiyittilla.
melpparanja parigananakal chuvadeyulla nirvachanathil mukalilekkulla adaykkal aavasyakathaye prachodippikkunnu "mathiyaya valuppam" ulla vasthukkal ellaayppozhum valuthakkaam. 
mattu rund vyavasthakal manasilakkaan, rolukal vipareethamaakkuka, pakaram kandethanulla oru "locating skeem" aayi pariganikkuka. eee vyaakhyaanathil, oraal chila sthalangalil tirayukayum lakshyam adangiyirikkunna aaa upavibhaagangale vivarikkan pratheekshikkukayum cheiyunnu. lakshyam evideyengilum aayirikkanam; athinaal shoonyamaaya settu orikkalum akathu kayaran kazhiyilla. rund upasettukalilum goal undengil, avayude pothu pradeshathekku "suum inn" cheyyanam. 
oro skeem gadakavum puthiya vivarangal nalkunna (onnukil "ivide nokkuka" allengil "mattevideyengilum nokkuka") oru "thikanja locating skeem" oru ultrafilter vivarikkunnu. oro tirayalum falapradamaanu", allengil mattoru vidhathil paranjaal, "oro locating skeem oru thirayal phalatthil avasaanikkunnu." 
chila topological spacinte "pothuvaaya" gadakangalaal thripthippedunna gunangal nirvachikkuka ennathaanu oru filterinte pothuvaaya upayogam. vyakthamaayi ezhuthaan prayaasamulla pooyintukal kandethaan eee aaplikkeshan "locating skeem" saamaanyavalkkarikkunnu.
nirvachanam.
bhaagikamaayi oordar cheytha settinte oru upavibhagam oru filter allengil duval ideal aanenkil
athum undengil, athu sariyaaya filter aanennu parayappedunnu. settu sidhaanthathileyum ganithashaasthra yukthiyileyum rachayithaakkalkk palappozhum ellah filterukalum sariyaayirikkanam; eee lekhanam aaa kanvenshan "ozhivaakkum". sariyaaya mattoru filttarilum adangiyittillaatha oru filteraanu ultrafilter. 
filter baseukal.
(i) srishtikkunna mukalile settu undengil athinte adisthaanamo adisthaanamo aanu oru upavibhagam.iii.ettavum cheria mukalilekku adanjirikkunna) ellaam. oro filter svayam oru aditharayaanu. 
koodaathe, shoonyavum thaazhekku nayikkappedunnathumaanel, oru filteraaya (athinu oru bays) oru mukalile settu srishtikkunnu. atharam settukale prefilterukal ennum mukalil paranja filter bays/bays ennum vilikkunnu, avaa srishtikkukayo vyaapikkukayo cheiyunnuvennu parayappedunnu. sariyaaya filter srishtikkukayaanenkil maathrame oru prefielter shariyaayulloo. 
nalkiyirikkunnathupole, settu adangiyirikkunna ettavum cheria filteraanu, chilappol ezhuthappettathum. atharamoru filterine principal filter ennu vilikkunnu; ithu pradhana ghadakam allengil srishtikkal ennu parayappedunnu. 
parishkkaranam.
rund prefilterukal oonaakkiyittundennu karuthuka, oronninum, atharathilulla oru undu. appol naam athu (allengil refines) aanennu parayunnu; athupole, (allengil parukkan) ennathinekkaal parukkan aanu. prefilterukalude settile oru preordaraanu refinement. vaasthavathil, avayum parishkarikkukayaanenkil, avaye thulyamennu vilikkunnu, kaaranam avaa ore filter srishtikkunnu. athinaal prefilteril ninnu filterilekkulla vazhi oru pree oordaringil ninnu anubandha bhagika oordaringilekku kadannupokunnathinte oru udaaharanamaanu.
prathyeka kesukal.
charitraparamaayi, ekapaksheeyamaaya bhagika oordarukalkku munbu oordar-theoritic laattisukalilekku filterukal saamaanyavalkkarikkappettu. laattisukalude kaaryatthil, thaazhekkulla disha parimithamaaya meettukalkk keezhil closser ennu ezhuthaam ellaavarkkum, oraalkk undu. 
leanier filterukal.
oru leanier (ultra) filter ennathu oru nischitha vekttar spacinte vekttar subspaceukalude laattisile oru (ultra) filteraanu, ulppeduthal kramatthil. vyakthamayum, oru vekttar speysile oru leanier filter vekttar subspaceukalude oru kudumbamaanu, athil vekttar subspace undengil, pinne pinne. 
athil adangiyittillengil oru leanier filter sariyaayirikkum.
oru settile filterukal; sub baseukal.
oru settu nalkiyaal, power settu bhaagikamaayi settu inclusion vazhi oordar cheiyunnu; eee posattile filterukale palappozhum "filterukal oonn" ennu vilikkunnu, padaprayogatthinte durupayogathil. atharam sthaanangalkku, thaazhekkulla dishayilum mukalilekkulla adachidalum kuraykkuka
sariyaaya/degenerate cheyyaatha filter adangiyittillaatha onnaanu, eee moonnu vyavasthakalum (degeneration allathava ulppede) henry kaarttante filterinte yadhaartha nirvachanamaan. settukalile filterukal sariyaayirikkanamennu aavashyappedunnathu-"saarvathrikamallenkilum"-saadhaaranamaanu (poost filterukalil oraalude nilapad enthuthanneyaayaalum); nammal veendum eee kanvenshan ozhivaakkum. 
oru settile prefilterukalil iva randum adangiyittillengil maathram avaa sariyaayirikkum. 
oro upavibhagathinum, athil adangiyirikkunna ettavum cheria filter undu. prefilterukaleppole, srishtikkukayo spanu cheiyukayo cheiyunnathaayi parayappedunnu; oru bays ennathu ellah parimithamaaya intersectionukaludeyum koottamaanu. shariyaayirikkumbol (athinaal) settu oru filter subbase aanennu parayappedunnu. 
settukalile sariyaaya filterukalkku parimithamaaya intersection property undu. 
anganeyaanengil, anuchithamaya filter maathram sweekarikkuka. 
soujanya filterukal.
oru filter athinte amgangalude intersection shoonyamaanengil athu swathanthramaanennu parayappedunnu. sariyaaya principal filter soujanyamalla. 
oru filterinte eathenkilum parimithamaaya amgangalude intersectionum oru amgamaayathinaal, oru parimitha settile sariyaaya filter soujanyamalla, mathramalla athinte ellah amgangaludeyum pothu intersection srishtikkunna pradhana filter ithaanu. ennaal ananthamaaya oru settile oru noonn principal filter soujanyamaayirikkanamennilla frechattu filter ulppeduthiyaal maathram oru filter soujanyamaanu (kaanuka).
udaaharanagal.
parimithamaaya sthaanathile filterukalude lalithamaaya udaaharanathinaayi eee lekhanathinte mukalilulla chithram kaanuka. 
bhagika kramatthil, pooyint thirichulla thaarathamyathiloode yadhaartha moolyamulla functionukalude idam. appol "ananthathayil valuth" enna functionukalude koottam, formula _ 1 oru filter aanu. domaine compactify cheythum codomaine poortthiyaakkiyum oraalkk eee nirmmaanam valare doore saamaanyavalkkarikkaan kazhiyum vyathirikthamaya upavibhagamulla oru settum vyathirikthamaya gadakamulla oru posittum aanenkil, oru filter inn aanu. 
settu oru filteraanu. kooduthal saadhaaranayaayi, eathenkilum directed settu aanenkil, formula _ 2 oru filteraanu, athine teyil filter ennu vilikkunnu. athupole eathoru nettum aakasmikatha filter srishtikkunnu. oru teyil filteraanu athinte saadhyataa filter. 
oru ananthamaaya settile frechattu filter isformula _ 3 oru alavedukkal sthalamaanengil, shekharanam oru filteraanu. appol athu oru filter aanenkil; frechattu filteraanu ennunna alavu. 
oru oordinal nalkiyaal, topology kramatthil adachittundengilum nettu-theoritic paridhi undengil oru upavibhaagathe club ennu vilikkunnu. clubbukal oru filter roopappeduthunnu club filter,. 
munbathe nirmmaanam inipparayunna reethiyil saamaanyavalkkarikkunnu eathoru clubb saandramaaya upavibhaagangalude (oordinal topologyil) oru shekharamaanu, koodaathe oro gadakavum paalikkunnu. idathoornna settukalude ekapaksheeyamaaya shekharam upayogichu maattisthaapikkunna, avide "saadhaaranayaayi" oro gadakatheyum kandumuttunna oru filter nilavilundu, athine oru generic filter ennu vilikkunnu. kanakkakkavunna vidhathil, rasiova-sikorsky lemma soochippikkunnathu atharamoru filter nilanilkkanam ennaanu; "cheria" ennaanaavaathathinaal, atharamoru filterinte nilanilpp maarttinte svayamsidhathayiloode nirbandhikkaam. 
parimithamaaya cardinalityude bhagika kramangalude koottamaaya modulo isomorphisathe soochippikkaam. bhaagikamaayi oordar cheyyuka 
appol aattomik ithara bhagika oordarukalude upavibhagam oru filter roopappeduthunnu. athupole, parimithamaaya cardinality, modulo isomorphisathinte chila commutative ringukalkku mukalilulla injective modyoolukalude koottamaanengil, oru bhagika kramam ithaanu 
ananthamaaya eathenkilum kardinal nalkiyaal, athile moduleukal oru filter roopeekarikkunna gadakangalil kurachumaathram srishtikkaan kazhiyilla. 
oru settile oro ekeekritha ghadanayum oru filter oonaakkunnu.
aadarshangalumaayulla bandham.
oru filterilekkulla eratta ashayam-athaayath, ellaam vipareethamaakkunnathiloodeya kaimaattam cheiyunnathiloodeyum labhikkuna ashayam-oru oordar idealaanu. eee dwaithatha kaaranam, filterukalude eathu chodyavum idealukalekkurichulla oru chodyathilekku yaanthrikamaayi vivarthanam cheyyaan kazhiyum, thirichum; prathyekichum, oru prime allengil maximal filter ennathu (yathaakramam) prime allengil maximal aaya oru filteraanu. 
anuyojyamaaya ideal kuranjathaanengil maathram filter oru ultrafilteraanu. 
model sidhaanthathil.
oru settile oro filterinum, formula _ 4 nirvachichirikkunna settu function parimithamayi additteev aanu-aaa padam ayanjathaayi kanakkaakkukayaanenkil oru "alavu". koodaathe, oru ultrafilteraanenkil angane nirmmicha alavukal ellaayidathum nirvachikkappedunnu. athinaal, stettment formula _ 5 eathaandu ellaayidathum nilanilkkunna prasthaavanaykk oru paridhivare samaanamaayi kanakkakkaam."ganithashaasthra yukthiyude oru shaakhayaaya model sidhaanthathile ultraprodect sidhaanthathil oru filterile amgathvathinte vyaakhyaanam (prachodanatthinaayi, yadhaarthamalla) upayogikkunnu.
topologyil.
pothuvaaya topologyilum vishakalanathilum, oru metrik speysile seekwansukalude punk pole kanverjansine nirvachikkaan filterukal upayogikkunnu. avaa vaividhyamaarnna ekapaksheeyamaaya topological edangaliludaneelam oru paridhi enna ashayathe ekeekarikkunnu. 
filterukalude aavasyakatha manasilakkaan, oru nettinte thulyamaaya aashayathil ninnu aarambikkuka. oru kramam saadhaaranayaayi swaabhaavika samkhyakalaal soochikayilaakkappedunnu, avaa poornnamaayum kramappeduthiya oru koottamaanu. nettukal oru ekapaksheeyamaaya directed settu upayogichu maattisthaapichu oru sequence enna ashayam saamaanyavalkkarikkunnu. aadyam kanakkakkavunna edangal polulla topological edangalude chila vibhagangalil, sequenceukal mikka topological gunangalum savisheshamaakkunnu, pakshe ithu pothuve shariyalla. ennirunnaalum, valakalum filterukalum ellaayppozhum aaa topological gunangale savisheshamaakkunnu. 
filterukalil topological spacinu purambhaagathulla oru settum ulppedunnilla, athesamayam sequenceukalum nettukalum mattu directed settukale aasrayikkunnu. ekkaaranathaal, ellah filterukaludeyum shekharam ellaayppozhum oru koottamaanu, athesamayam ellah moolyamulla valakaludeyum shekharam sariyaaya oru vibhagamaanu.
ayalpakka thaavalangal.
topological sthalathe eathu pooyintum oru ayal filtero sistamo nirvachikkunnu athaayath, avayude inteeriyaril adangiyirikkunna ellah settukaludeyum kudumbam. oru koottam ayalpakkangal oru ayalpakka aditharayaanu. thulyamaayi, anganeyonnu undengil maathram athinte oru ayalpakkamaan. 
kanverjant filterukalum klasttar pooyintukalum.
ayalpakka filter adangiya oru filter srishtikkukayaanenkil maathram oru prefielter oru poyintilekku othucherunnu-vyakthamaayi, oro ayalpakkathinum, atharam chilath undu. kuranja vyakthamaayi, parishkkarikkukayaanenkil maathram, eee avasthayil eathenkilum ayalpakka aditharaykku pakaram vaykkaan kazhiyum. vyakthamayum, oro ayalpakka aditharayum othucherunnu. 
oru filter (svayam srishtikkunna) iff aayi othucherunnu. ayal filterinte swabhaavasavisheshathakal vyakthamakkaan mukalil paranjava thirichedukkaam oro filter samyojippikkunnathinekkal mikacha filter parukkanaan. 
engil, athine paridhi (pooyint) ennu vilikkunnu. oro gadakathinum oro ayalpakkavumaayi shoonyamallaatha intersection undengil maathram prefilter klasttar cheiyumennu (allengil oru klasttar pooyintaayi) parayappedunnu. oro limitt pooyintum oru klasttar pooyintaanengilum neremarichu pothuve shariyalla. ennirunnaalum, oru filterinte oro klasttar pooyintum oru limitt poyintaanu.

