afat gut

ganitat, eka vipul gut haah eka topological gut aahe joou eka vishisht arthane maryadit gatanchya pranalitoon ekatra keela jaato. 
maryadit gatanchya sampoorn pranaliche 'eksaman' kinwa 'synoptic' drushya pradan karane hii eka vistrut gut vaparanyachi kalpana aahe. profinite groupche gunadharm samanyatah pranaliche eksaman gunadharm aahet. udaharanarth, jarr aanee keval sootra 1 astitvat asel tarr pranalitil pratyek gut sootra _ 2 ghatkandware tayar keela jaau shakato, tarr haah pradirgh gut maryaditpane (topological gut mhanoon) tayar keela jaato. maryadit gatambaddalchya anek prameyanna sahajpane vistrut gatanmadhye samanyikrut kele jaau shakate; udaharanadakhal lagrangeche pramey aanee sayalo pramey aahet.
eka vistrut gut tayar karnyasathi ekhadyala maryadit gatanchi pranali aanee tyanchyamadhye samooh samarupata aavashyak asate. samanyata naan gamavata, heey samarupata heey surjective asalyache gruheet dharale jaau shakate, asha paristhitit maryadit gut parinami prophenite gatache bhagakar gut mhanoon disatil; eka arthane, heey bhagakar prophenite gatacha andaje andaj lavtat. 
3-edic purnankanche yogatmak gut aanee anant-degree kshetra vistaranche gallois gut hii profinite gatanchi mahattvachi udaharane aahet.
pratyek vishal gut sankshipt aanee purnapane disconnect kelela asato. sankalpaneche sankshipt nasalele samanyikaran heey sthanik patalivarchya alpasankhyak gatanche aahe. purnapane disconnect kelele gut heey aanakhi samanya aahet.
vyakhya.
vishal gatanchi vyakhya done samatulya marganpaiki konatyahi eka prakare kelee jaau shakate.
pahili vyakhya (rachnatmak).
eka prophenite gut haah eka topological gut aahe joou swatantra maryadit gatanchya inverse pranalichya inverse maryadeshi samaroop aahe. yaa sandarbhat, eka ulat pranalimadhye eka nirdeshit sanch sootra 4, eka maryadit gatanche anukramit kutumb sootra 5, pratyekamadhye swatantra topology asate aanee samarupata sootra 6 asey asate qee sootra 7 heey sootra 8 vareel olakh nakasha aahe aanee sangrah rachana gunadharm sootra 9 chey samadhan karato, ulat maryada haah sanch aaheh
sootra _ 10
sapeksh utpadan topologies susajj.
sarvatrik malmattechya sandarbhat ulat maryada dekhil paribhashit kelee jaau shakate. spasht shabdat, heey saha-filter kelelya maryadechya bandhkamache eka vishesh prakaran aahe.
dusari vyakhya (svayamsiddha).
eka vishal gut haah eka bhoo-aachchadan, sankshipt aanee purnapane disconnect kelela topological gut aaheh mhanje, eka topological gut joou eka dagadi jaaga dekhil aahe.
apoorn purnata.
eka aniyantrit gut sootra dile tarr sootra 13 chey eka sambandhit prophenite gut sootra aahe, teey gut sootra 14 chee inverse maryada mhanoon paribhashit kele jaate, jethe sootra 15 heey maryadit nirdeshankachya sootra 16 madheel samanya upasamuhanmadhun jaate (heey samanya upasamuh anshatah samaveshadvare krambaddha kele jaatat, jeey bhagandarmyan naisargik samarupatechya inverse pranalimadhye rupantarit hote).
eka naisargik samarupata sootra aahe _ 17 aanee yaa samarupata antargat sootra _ 16 chee pratima sootra _ 19 madhye ghan aahe, samarupata sootra _ 20 heey jarr aanee fakt jarr gut sootra _ 16 avashisht maryadit asel (i.eae.,
sootra _ 22 jethe chhed maryadit nirdeshankatil serve samanya upasamuhanmadhun jaato).
samarupata sootra _ 20 heey khaleel sarvatrik gunadharmandware darshavile jaateh konatehi anant gut sootra _ 24 aanee konatehi satat gut samarupata sootra _ 25 dile aahe jethe sootra _ 16 laa gut karyanshi susangat sarvat lahan topology dilee jaate jyamadhye tyache maryadit nirdeshankache samanya upasamuh ughade asatat, tethe sootra _ 28 saha eka adviteey nirantar gut samarupata sootra _ 27 astitvat aahe.
samanata.
pahilya vyakhyedvare tayar kelela konatahi gut yaulat vyakhyetil svayamsiddhateche samadhan karato.
yaulat, yaulat vyakhyetil svayamsiddha tattvanche samadhan karnare konatehi gut sootra _ 16 heey pahilya vyakhyenusar, inverse limit sootra _ 30 vaparoon inverse limit mhanoon tayar kele jaau shakate, jethe sootra _ 15 heey (ulat) samaveshadvare krambaddha kelelya sootra _ 16 chya khulya samanya upasamuhanmadhun jaate. jarr sootra _ 16 heey topologicaldrushtya maryaditpane tayar kele gele asel tarr teey tyachya swatachya afat purnatechya barobariche aahe.
surjective pranali.
vyavharat, maryadit gatanchi ulat pranali jawaljawal nehmich asate, yacha arth tyache serve nakashe surjective asatat. samanyata naan gamavata, keval surjective pranalincha vichar karane purese aahe kaaran konatihi inverse pranali dilyas, pratham tyache prophenite group formula _ 11 tayar karane aanee nantar teey swatache prophenite completion mhanoon tayar karane shakya aahe.
anant gut.
eka kalpana aahe, jee sankalpanatmak duheri teey anant gut aahe; i.eae. jarr tee maryadit gatanchya prerak pranalichi thet maryada asel tarr gut sootra _ 16 heey anant aahe. (visheshtah, haah eka ind-gut aahe.) nehmichi shabdavali vegali aaheh jarr pratyek maryaditpane tayar kelela upasamuh maryadit asel tarr gut sootra _ 16 laa sthanik patalivar maryadit mhanatat. heey kharey tarr 'anant' asanyaitkech aahe.
pontriagin dwaitata lagoo karoon, ekhadyala asey disun yete qee abelian prophenite gut sthanik patalivar maryadit swatantra abelian gatansah dwait aahet. nantarche fakt abelian tortion gut aahet.
prakshepak vishal gut.
jarr pratyek vistarasathi uchalanyachi malmatta asel tarr eka vishal gut asato. heey asey mhananyaitakech aahe qee sootra _ 16 prakshepak aahe jarr eka aseem sootra _ 89 madheel pratyek surjective morphismsathi eka vibhag sootra _ 90 asel
_ 16 yaa nikhalas gut sutrasathi prakshepakata hii donpaiki konatyahi eka gunadharmanchya samtulyah
pratyek prakshepak prophenite gut haah chhadm bijganitiyadrushtya bund kshetracha eka paripoorn gallois gut mhanoon olakhala jaau shakato. alekzandar lubotzki aanee lau vanne dayne korade padalyamule haah parinam hoto.
procyclic gut.
eka prophenite group formula _ 16 mhanje jarr teey ekach ghatak sutradvare topologicaldrushtya tayar kele gele asel tarr _ 98 mhanje, jarr sootra _ 99 asel tarr upasamuh sootra _ 100 chey samapan
jarr aanee keval sootra 102 jethe sootra _ 3 serve abhajya sankhyanvar shrenibaddha asel aanee sootra _ 104 heey sootra _ 36 kinwa sootra _ 106 sathee samaroop asel, tarr topological gut sootra _ 16 heey procyclic aahe.

