avkash (ganeet)

ganitat, jaaga haah kahee atirikt rachana asalela sanch (kadhikadhi vishva mhanatat) asato.
aadhunik ganitat euclidean jaaga, resheey jaaga, topological jaaga, hilbert jaaga kinwa sambhavyata jaaga yasarkhya anek prakarchya jaaga vaparlya jaat asalya taree, tyaa swatah "jaaga" yaa kalpanechi vyakhya karat naheet.
jaaget nivadak ganiteey vastoo asatat jyanna bindu manale jaate aanee yaa bindunmadhil nivadalele sambandh asatat. 
bindunche swaroop mothya pramanat badaloo shakateh udaharanarth, bindu heey sanchache ghatak, kinwa jagevaril karye kinwa kinwa jaageche upasthan asoo shakatat. heey natesambandh aahet jeey jaageche swaroop paribhashit karatat. adhik tantotant, samaroop jaaga eksarkhya manalya jaatat, jethe done jaganmadhil samarupata mhanje tyanchya bindunmadhil eka-teey-eka sanvad asato joou sambandh jatan karato. udaharanarth, tri-aayami euclidean jagechya bindunmadhil sambandh heey yuklidchya svayamsiddhatedvare advitiyapane nirdharit kele jaatat aanee serve tri-aayami euclidean jaaga samaan manalya jaatat.
satatyasarkhya topological kalpananchya pratyek euclidean jaaget naisargik vyakhya asatat. 
tathapi, topology vakr reshanpasoon saral reshanmadhye farak karat nahee aanee tyamule euclidean aanee topological spacemadhil sambandh 'vismaraneey' aahe. yaa prakarche sambandh 'spaceche prakar' yaa vibhagat adhik tapshilvar hatalale jaatat. 
dilelya ganiteey vastoola bhaumitik "jaaga" kinwa bijganitiy "rachana" manale pahije qee nahee heey nehmich spasht hott nahee. borbakine prastavit keleli 'rachana' chee samanya vyakhya, serve samanya prakarchya jaaga samavisht karate, samarupatechi samanya vyakhya pradan karate aanee samaroop sanrachananmadhil gunadharmanchya hastantaranas nyayya tharavate.
itihas.
bhoomitichya suvarnayugachya aadhi.
pracheen greek ganitat, 'avkash' haah dainandin jeevanat jeevanat trimiteey vastavacha bhaumitik amoort bhag hota. ii. sha. puu. 300 chya sumaras, euclidne jagechya gunadharmansathi svayamsiddha tattve dilee. euclidne yaa bhaumitik payanvar serve ganeet tayar kele, joparyant ola vibhaganchya lambichi tulana nivadalelya sandarbh vibhagachya lambishi karoon sankhya paribhashit karnyasathi jaat aahe.
1637 madhye rene descartes yanni nirdeshankanchi (vishleshanatmak bhoomiti) paddhat sweekarli. tyaveli, bhaumitik prameyanna naisargik vidnyanachya vastunpramanech antardnyan aanee tarkadvare dnyat honare paripoorn vastunishtha satya manale jaat hote; aanee svayamsiddha shabdanna vyakhyanche spasht parinam manale jaat hote.
bhaumitik aakadyanmadhil done samatulya sambandh vaparale geleh ekarupata aanee samanata. anuvad, paribhraman aanee pratibimb aakrutiche ekroop aakrutinmadhye rupantar karatat; samarupata-samaan aakrutinmadhye. udaharanarth, serve vartule paraspar samaan aahet, parantu lambvartul vartulansarkhi naheet. 1795 madhye gaspard mongane sadar kelela tisra samatulya sambandh prakshepak bhoomitimadhye aadhalatoh keval lambvartulach navhe tarr paravalay aanee atidhruviy, yogya prakshepak parivartanankhali vartulanmadhye rupantarit hotat; teey serve prakshepak samatulya aakruti aahet.
euclidean aanee projective yaa done bhoomitimadhil sambandh darshavito qee ganiteey vastoo aaplyala tyanchya sanrachanesah dilya jaat naheet. tyaeivaji, pratyek ganiteey siddhant tyachya vastunche varnan tyanchya kahee gunadharmandware karato, tantotant tyaa jyaa siddhantachya payavar swayamsiddhi mhanoon thevalya jaatat.
prakshepak bhoomitichya prameyanmadhye antar aanee coone disu shakat naheet, kaaran yaa kalpana prakshepak bhoomitichya svayamsiddhanmadhye namood kelya jaat naheet kinwa tethe namood kelelya kalpananmadhun paribhashit kelya jaat naheet. trikonachya teen konanchi berries kitee aahe" haah prashna euclidean bhoomitimadhye arthpurn aahe parantu prakshepak bhoomitimadhye arthaheen aahe.
aahe shatkat eka vegali paristhiti udbhavalih kahee bhoomitimadhye trikonachya teen konanchi berries changlya prakare paribhashit aahe parantu shastriy mulyapeksha (180 ansh) vegali aahe. 1829 madhye nikolai lobachevsky aanee 1832 madhye janos bolay yanni (aanee 1816 madhye carl friedrich goss, aprakashit) sadar keleli gair-euclidean hyperbolic bhoomitine mhatale aahe qee berries trikonavar avalambun asate aanee nehmich 180 anshanpeksha qamee asate. 1868 madhye eugenio beltramy aanee 1871 madhye felix clain yanni gair-euclidean hyperbolic bhoomitiche euclidean "models" midavale aanee tyadvare yaa siddhantala tarkik shakyata mhanoon purnapane nyayya tharavale.
yaa shodhamule euclidean bhoomitichya paripoorn satyacha dhong sodoon denyas bhag padale gele. tyat asey disun aale qee swayamsiddhi "spasht" kinwa "vyakhyanche parinam" naheet. tyaeivaji, teey gruhitak aahet. teey prayogik vastavashi kitee pramanat julatat? yaa mahattvachya bhautik samasyecha aata ganitashi kahihi sambandh nahee. jaree 'bhoomiti' prayogik vastavashi julat nasali taree, tyache pramey 'ganiteey satya' peksha qamee naheet.
gair-euclidean bhoomitiche euclidean model mhanje euclidean jaaget astitvat asalelya kahee vastunchi nivad aanee yaa vastunmadhil kahee sambandh jeey gair-euclidean bhoomitichya serve svayamsiddhateche (aanee mhanoon, serve pramey) samadhan karatat. yaa euclidean vastoo aanee sambandh, samkalin kalakaranpramane, eka pracheen sadarikaran karat, gair-euclidean bhoomitichi "bhoomika" karatat. pratyakshat kadhihi naan ghadalelya paristhitiche anukaran kalakar karoo shakatat. rangmanchavaril kalakaranmadhil sambandh heey natkatil patranmadhil sambandhanche anukaran karatat. tyachpramane, euclidean pratikrutichya nivadalelya vastunmadhil nivadalele sambandh heey gair-euclidean sambandhanche anukaran karatat. heey darshavate qee ganitat vastunmadhil sambandh aavashyak aahet, tarr vastunche swaroop nahee.
bhoomitiche suvarnayug aanee tyanantarche.
bhoomiti" yaa shabdacha (pracheen greekah bhoo-"pruthvi",-meteron "mapan") suruvatila arth aapan jyaa jaaget rahato tyaa jaaget lambi, pradesh aanee khandanvar prakriya karnyacha eka vyavharik marg hota, parantu nantar tyacha mothya pramanat vistar karnyat aala (tasech yethe jaagechi kalpana).
borbakichya matey, 1795 (mongche "geometry discriptive") aanee 1872 (claincha "erlangen program") yaa darmyanchya kalala "bhoomitiche suvarn yug" mhatale jaau shakate. euclidne tapasalelya mool jagela aata trimiteey euclidean jaaga mhanatat. 23 shatakanpurvi euclidne suru kelelya tyachya svayamsiddhatemadhye hilbertchya svayamsiddhatesaha, tarskeechya svayamsiddhatesaha aanee berkhoffchya svayamsiddhatesaha sudharana karnyat aalee. yaa svayamsiddha pranali anek svayamsiddha tattvandvare maryadit kelelya adim kalpanandware (jase qee "bindu", "darmyan", "sarvasamaveshak") jaageche varnan karatat.
vishleshanatmak bhoomitine mothee pragati kelee aanee parivartan gatanchya aparivartanandvare shastriy bhoomitichya prameyanna gananesah badalanyat yash midavale. tevhapasoon, shastriy bhoomitiche naveen pramey vyavsayik ganitshastradnyanpeksha haushi lokansathi adhik swarasyapoorn rahile aahet. tathapi, shastriy bhoomiticha varsa haravala nahee. borbakichya matey, "eka swayatt aanee jiwant vidnyan mhanoon tyachya bhoomiketoon pudhe gelyanantar, shastriy bhoomitiche asha prakare samkalin ganitachya sarvatrik bhashet rupantar jhaale aahe".
ekach velee, ganitacha paya mhanoon sankhyanni bhoomitiche sthanantar karnyas suruvat kelee. udaharanarth, richard dedekindchya 1872 chya nibandh 'statigkit und irrational zahlen' (satatya aanee atarkik sankhya) madhye, teey thampane sangatat qee reshevaril bindunmadhye dedeckind katche gunadharm asavet aanee mhanoon eka resha hii vastavik sankhyanchya sanchasarkhich hotee. heey siddha houu naan shakanare eka gruhitak aahe heey lakshat ghenyakarta dedeckind savadh aahe. aadhunik upcharanmadhye, dedekindcha dawa hii bahutekda reshechi vyakhya manali jaate, jyamule bhoomiti ankganitashi qamee hote. tri-aayami euclidean jaaga hii eka sanlagn jaaga mhanoon paribhashit kelee jaate, jichya ghatakanchya farkanchya sambandhit vector jaaga aateel utpadanasah susajj aahe. euclidmadhye mhatalyapramane, 'survatipasun' hii vyakhya aata sahasa vaparali jaat nahee, kaaran tee yaa jaagecha itar jaganshi asalela sambandh ughad karat nahee. tasech, tri-aayami prakshepak jaagechi vyakhya aata chaar-aayami vector jagechya serve eka-aayami upasthananchi (mhanje moodatoon saral resha) jaaga mhanoon kelee jaate. payabharanitil yaa badalasathi svayamsiddha tattvancha eka naveen sanch aavashyak aahe aanee jarr heey svayamsiddha tattva sweekarle gele tarr bhoomitiche shastriy svayamsiddha tatva pramey banatat.
aata jaaget nivadak ganiteey vastoo (udaharanarth, kinwa jagevaril karye, kinwa kinwa jaagechi upasthanke, kinwa fakt eka sanchache ghatak) asatat, jyanna bindu manale jaate aanee yaa bindunmadhil nivadalele sambandh asatat. tyamule, jaaga yaa keval soyichya ganiteey rachana aahet. itar ganiteey vastunpeksha 'space' navachya sanrachana adhik bhaumitikdrushtya samajalya jaatil ashee apeksha kelee jaau shakate, parantu heey nehmich kharey nasate.
1854 madhye bernhard remanne dilelya prasiddha udghatan vyakhyananusar, "aine" vastavik sankhyanni parameters kelelya pratyek ganiteey vastoola asha serve vastoonchya "aine"-aayami jaagecha bindu manale jaau shakate. samkalin ganitshastradnya yaa kalpaneche niyamitpane anusaran karatat aanee jawaljawal sarvatra shastriy bhoomitichi shabdavali vaparane atyant soochak aahe asey tyanna vatate.
karye hii ganiteey mahattvachi vastoo aahet. remanne aadhich namood kelyapramane aanee 20 vya shatkat karyatmak vishleshanadvare vistrut kelyapramane, samanyatah teey anant-aayami karyasthale tayar karatat.
jaganche vargikaran.
teen vargikaran shreni.
pratyek prakarchya jaagechi swatachi vyakhya asali taree, 'jaaga' chee samanya kalpana aupcharikaran talhate. kahee sanrachananna jaaga mhanatat, tarr itar aupcharik nikashanshivay nasatat. shivay, 'rachana' chya sarvasadharan kalpanevar ekmat nahee.
pudalakchya matey, "ganeet [...] ganiteey sanrachanesarkhya ekach sankalpnene purnapane spasht kele jaau shakat nahee. tarisuddha, borbakicha sanrachanavadi drushtikon haah aaplyatala sarvottam aahe."
aata aapan borbakichya bhavnene jaganchya (aanee sanrachananchya) sambhavya vargikaranachi ruparesha aakhat asatana, aapan shevatchya vibhagat 'jaaga aanee sanrachana' madhye borbakichya sanrachanavadi drushtikonakade parat jaau.
aapan teen staranvar jaganche vargikaran karato. pratyek ganiteey siddhant tyachya vastunche varnan tyanchya kahee gunadharmandware karato heey lakshat gheta, vicharayacha pahila prashna aaheh konate gunadharm? yamule pahilya (varchya) vargikaran pataliparyant pohochate. kinwa staravar, visheshtah mahattvachya prashnanchi uttare vicharat ghetali jaatat (pahilya staranusar arthpurn asalelya prashnanpaiki). vargikaranachya vargikaranachya staravar, serve sambhavya prashnanchi uttare vicharat ghetali jaatat.
udaharanarth, "uchch-stariy vargikaran" heey euclidean aanee prakshepak jaganmadhye farak karate, kaaran done bindunmadhil antar euclidean jaganmadhye paribhashit kele jaate parantu prakshepak jaganmadhye paribhashit kele jaat nahee. 
aanakhi eka udaharan. trikonachya teen konanchi berries kitee aahe" haah prashna euclidean jaaget arthpurn aahe parantu prakshepak jaaget nahee. gair-euclidean jaaget yaa prashnas arth aahe parantu tyache uttar vegale dile jaate, jeey uchch-stariy farak nahee.
tasech, euclidean pratal aanee euclidean trimiteey jagemadhil farak haah uchch-stariy farak nahee; donhi prakarananmadhye 'pariman kaay aahe' haah prashna arthpurn aahe.
dviteey-stariy vargikaran", udaharanarth, euclidean aanee gair-euclidean jaganmadhye farak karate; maryadit-aayami aanee anant-aayami jaganmadhye; sankshipt aanee bigger-sankshipt jaganmadhye, ii.
borbakichya drushtine, kinwa staravaril vargikaran mhanje "prajati" dware vargikaran. jaivik vargikaranapeksha vegali jaaga anek prajatinchi asoo shakate.
udaharanarth, "truteey-stariy vargikaran" vegvegalya parimananchya jaganmadhye farak karate, parantu dvi-aayami euclidean jaaga mhanoon manalya olakhalya tri-aayami euclidean jagechya pratalamadhye aanee dvi-aayami euclidean jaaga mhanoon manalya olakhalya vastavik sankhyanchya serve jodyanchya sanchamadhye farak karat nahee. tyachpramane teey samaan gair-euclidean jagechya vegvegalya euclidean modelmadhye farak karat nahee.
adhik aupcharikpane, tisra star samarupateparyantchya jaganche vargikaran karato. done jaganmadhil samarupata hii pahilya jaageche bindu aanee kinwa jaageche bindu yanchyatil eka-teey-eka sanvad mhanoon paribhashit kelee jaate, jee pahilya staranusar nirdharit kelele serve sambandh jatan karate. paraspar samaroop jaaga ekach jagechya pratee mhanoon manalya jaatat. jarr tyapaiki eka dilelya prajaticha asel tarr teey serve karatat.
samarupatechi kalpana uchch-stariy vargikaranavar prakash takte. ekach uchch-stariy vargachya done jaganmadhil eka-eka patravyavahar dilyas, too samarupata aahe qee nahee heey konihi vicharoo shakate. vegvegalya varganchya done jagansathi yaa prashnala kahee arth nahee.
samarupatela swatahsathi svayamchalitataa mhanatat. euclidean jaageche automorphism mhanje badal, paribhraman, pratibimb aanee tyanchi rachana. euclidean jaaga yaa arthane eksandh aahe qee pratyek bindu kahee automorphismdware pratyek itar bindumadhye rupantarit keela jaau shakato.
euclidean svayamsiddha sootra konatehi swatantrya sodat naheet; teey jaageche serve bhaumitik gunadharm advitiyapane nirdharit karatat. adhik achookpaneh serve trimiteey euclidean jaaga paraspar samaroop aahet. yaa arthane aaplyakade tri-aayami euclidean jaaga aahe. borbakichya shabdat, sambandhit siddhant "atulaneey" aahe. yaulat, topological spaces samanyatah gair-samaroop asatat; tyancha siddhant "bahugunit" aahe. ganiteey tarkshastrat ashich kalpana udbhavateh jarr ekach cardinalityche serve model paraspar samaroop asateel tarr siddhantala shrenibaddha mhanatat. borbakichya matey, bahuayami siddhantancha abhyas heey sarvat lakshavedhi vaishishtya aahe jeey aadhunik ganitala shastriy ganitapasoon vegale karate.
jaganchya prajatinmadhil sambandh.
topological kalpana (satatya, abhisaran, khule sanch, bund sanch ii.) pratyek euclidean jaaget naisargikaritya paribhashit kele jaatat. kinwa shabdant, pratyek euclidean jaaga dekhil eka topological jaaga aahe. done euclidean jaganmadhil pratyek samarupata hii sambandhit topological jaganmadhil samarupata dekhil aahe (jyala 'homomorphism' mhanatat), parantu ulat chukiche aaheh samarupata antar vikrut karoo shakate. borbakichya shabdat, "topological space" hii "euclidean space" sanrachanechi "antarnihit" rachana aahe. shreni siddhantat ashach kalpana udbhavtatah euclidean jaganchi shreni hii topological jaganchya shrenivaril eka thos shreni aahe; visarnara (kinwa "kadhoon taknara") functor aadhichya shrenicha nantarchya shrenimadhye nakasha banavato.
tri-aayami euclidean jaaga heey euclidean jaageche eka vishesh udaharan aahe. borbakichya drushtine, tri-aayami euclidean antaralatil prajati euclidean antaralatil prajatinpeksha "samruddha" aahet. tyachpramane, sankshipt topological jaagechi prajati topological jagechya prajatinpeksha adhik samruddha aahe.
anjirmadhye dakhavalyapramane avkashanchya prajatinmadhil asey sambandh aakrutidvare vyakt kele jaau shakatat. 3. aaa teey buub paryantchya banancha arth assa aahe qee pratyek buub dekhil aaa aahe kinwa tyala aaa mhanoon manale jaau shakate kinwa aaa ityadi pradan kele jaau shakate. aaa aanee buub yanna jaganche varg mhanoon vicharat ghetalyas banala aaa teey buub asey sankraman mhanoon arth lavala jaau shakato. (borbakichya shabdat, aaa, buub varg sanch naslyas aaa yaa karyatoon aaa chee "vaja karanyachi prakriya"; haah sukshmabhed khaleel goshti avaidh karat nahee.) anjiravar done baana. 3 ulat karata yet naheet, parantu vegvegalya karananmule. 
euclidean' pasoon 'topological' kadey honare sankraman vismaraneey aahe. topology satat aanee visangat vegale karate, parantu lambaresh aanee vakrareshanmadhye farak karat nahee. antardnyan aaplyala sangate qee euclidean rachana topologymadhun punarsanchayit kelee jaau shakat nahee. purava topological jaageche automorphism (mhanje self-homeomorphism) vaparato jeey euclidean jaageche automorphism nahee (mhanje, shift, rotation aanee reflectionchi rachana nahee). asha prakarche parivartan dilelya euclidean sanrachaneche (samaroop parantu) bhinn euclidean sanrachanet rupantar karate; donhi euclidean sanrachana ekach topological sanrachaneshi sambandhit aahet.
yaulat, '3-diim euclidean' pasoon 'euclidean' madhye honare sankraman visarnyasarkhe nahee; euclidean jaaga 3-aayami asanyachi garaj nahee, parantu jarr tee 3-aayami asel tarr tee purn vikasit aahe, konatihi rachana nasht hott nahee. kinwa shabdant, nantarche sankraman heey injection (eka-eka) asate, tarr pahile sankraman injection (anek-eka) nasate. aapan kateri shepatichya banadvare injective sankraman darshavito, "′" aivaji "′".
donhi sankraman heey surjective nasatat, mhanje pratyek bii-space kahee aii-spacemadhun parinam dett nahee. pratham, 3-diim euclidean jaaga hii euclidean jaagechi eka vishesh (samanya nahee) cases aahe. dusare mhanje, euclidean jaageche topology heey topologyche eka vishesh prakaran aahe (udaharanarth, teey compact nasalele aanee jodalele asane aavashyak aahe, ii.). aapan '′ aivaji' ′ 'yaa done dokyanchya banadvare surjective sankraman darshavito. udaharanarth anjir pahaa. 4; tithe, "vastavik rekheey topological" teey "vastavik rekheey" assa baana done-dokyancha asato, kaaran pratyek vastavik rekheey jaaga tyachya rekheey sanrachaneshi susangat asalelya kahee (kiman eka) topologyla swikarte.
ashee topology sarvasadharanpane adviteey nasate, parantu jevha vastavik resheey jaaga maryadit-aayami asate tevha adviteey asate. yaa jagansathi sankraman heey injection aanee surjective donhi aahe, mhanje bijective; ankivar baana 'maryadit-kshitij vastavik rekheey topological' pasoon 'maryadit-kshitij vastavik rekheey' paryant pahaa. 4. ulat sankraman astitvat aahe (aanee kinwa, mage asalelya banadvare dakhavale jaau shakate). asha prakare sanrachananchya done prajati samaan aahet. vyavharat, sanrachana samatulya prajatinmadhye konatahi farak keela jaat nahee. anjiravaril mothya petine darshavilyapramane samatulya sanrachana ekach rachana mhanoon manalya jaau shakatat. 4.
bananni darshavilele sankraman samarupateche palan karatat. mhanje, done samarupi done samarupikade netat.
anjiravaril aakruti. 4 heey parivartaneey aahe. mhanjech, aakrutitil serve nirdeshit marg, jyachi suruvat aanee shevat samaan aahet, teey samaan parinamankade netat. anjiravaril desh kelele baana vagalata, khaleel itar aakruti dekhil parivartaneey aahet. 9. tithe spasht kelelya karanasathi 'topological' teey 'mojanyayogya' yaa banala desh kele jaateh 'topological jaageche mojanyayogya jaaget rupantar karnyasathi ekhadyane tyala σ-bijganit dile aahe. borel sanchanche σ-bijganit haah sarvat lokpriy aahe, parantu ekmev paryay nahee."eka ghan baana eka prachalit, tathakathit" "vihit" "sankraman darshavito jeey naisargikaritya swatahla soochit karate aanee mothya pramanat vaparale jaate, bahutekda purvanirdharitapane". udaharanarth, euclidean jaaget satatchya karyabaddal bolayache jhalyas, ekhadyala tyache topology spashtapane nirdisht karanyachi aavashyakta nahee. kharey tarr, paryayi topology astitvat aahet aanee kahivela vaparlya jaatat, udaharanarth, bareek topology; parantu yaa nehmi spashtapane nirdisht kelya jaatat, kaaran tyaa prachalit topologypeksha khoopach qamee ullekhaneey aahet. eka desh kelela baana soochit karato qee anek sankraman vaparat aahet aanee konihi farsa prachalit nahee.
jaganche prakar.
resheey aanee topological jaaga.
done moolbhut jaaga mhanje resheey jaaga (jyala vector jaaga dekhil mhanatat) aanee topological jaaga.
resheey jaaga bijganitiy swarupachya asatat; vastavik resheey jaaga aahet (vastavik sankhyanchya kshetravar),
jatil resheey jaaga (jatil sankhyanchya kshetravar) aanee adhik samanyatah, konatyahi kshetravaril resheey jaaga. pratyek jatil resheey jaaga hii eka vastavik resheey jaaga dekhil asate (nantarchi 'aadhichi' aahe), kaaran pratyek jatil sankhya done vastavik sankhyanni nirdisht kelee jaau shakate. udaharanarth, eka-aayami jatil resheey jaaga mhanoon manale jaanare jatil pratal dvi-aayami vastavik resheey jaaget qamee kele jaau shakate. yaulat, vastavik resha hii eka-aayami vastavik resheey jaaga mhanoon manali jaau shakate parantu jatil resheey jaaga mhanoon nahee. kshetra vistar dekhil pahaa. adhik samanyatah, kshetravaril vector spacemadhye tyaa kshetrachya upkshetravaril vector spacemadhil rachana dekhil asate.
vyakhyenusar resheey jaaget dilelya resheey karyanmule, saral resha (aanee samtal aanee itar resheey upkshetra); samantar resha; lambvartul (aanee lambvartulakar) asha kalpana nirman hotat. tathapi, paribhashit karane ashakya aahe 
orthogonal (lumb) resha, kinwa lambvartulanmadhil vartule ekeri karane, kaaran resheey jaaget 
coone mojanyasathi vaparale jaau shakanare scalar utpadan ashee konatihi rachana nahee. resheey jaageche pariman heey resheey swatantra vahakanchi kamal sankhya mhanoon kinwa, samatulya, jaaga jaaga vahakanchi kiman sankhya mhanoon paribhashit kele jaate; teey maryadit kinwa anant asoo shakate. ekach kshetravaril done resheey jaaga jarr aanee keval jarr tyaa samaan parimanachya asateel tarach samaroop asatat. eka jatil resheey jaaga hii dekhil eka vastavik resheey jaaga aahe.
topological jaaga vishleshanatmak swarupachya asatat. vyakhyenusar topological spacemadhye dilele khule sanch, satat karye, marg, nakashe; abhisaran anukram, maryada; aateel, seema, bahya asha kalpanankade netat. tathapi, eksaman satatya, baddh sanch, kachee anukram, bhinn karnyayogya karye (marg, nakashe) hii anirdharit rahatat. topological spacemadhil samarupatela paramparikpane samarupata mhanatat; heey donhi dishanna satat eka-eka sanvad aahet. ughade antar (0,1) sampoorn vastavik reshesathi (− φ, φ) samaroop aahe parantu bund antarasathi [0,1] samaroop nahee, kinwa vartulasathi nahee. ghancha prushthbhag golasathi (chenducha prushthbhag) samaroop asato parantu torresathi samaroop nasato. vegvegalya parimananchya euclidean jaaga homomorphic nasatat, jeey spasht disate, parantu siddha karane sope nahee. topological jaageche pariman paribhashit karane katheen aahe; prerak pariman (bhaumitik aakrutichya simeche pariman samanyatah aakrutichya parimanapeksha eka qamee asate yaa nirikshanavar aadharit) aanee lebsg covering pariman vaparale jaau shakate. euclidean jagechya babatit, donhi topological parimane "aine" chya barobarichi asatat.
topological jaagecha pratyek upasanch swatach eka topological jaaga aahe (yaulat, resheey jaageche keval "resheey" upasanch resheey jaaga asatat). samanya topologydware tapasalya gelelya (jyala point-seet topology dekhil mhanatat) aniyantrit topological jaaga, homomorphismparyantchya sampoorn vargikaranasathi khoop vaividhyapurn aahet. sankshipt topological jaaga haah topological jagancha eka mahattvacha varg aahe (yaa "prakarchya" prajati). pratyek nirantar karya asha jagevar bandhane ghatalele asate. bund madhyantar [0,1] aanee vistarit vastavik resha [− φ, φ] sankshipt aahet; ughade madhyantar (0,1) aanee resha (− φ, φ) naheet. bhaumitik topology anek patancha (yaa "prakarchya" aanakhi eka "prajati") tapas karate; hii sthanik patalivar euclidean jagansathi samaroop topological jaaga aahet (aanee kahee atirikt atty purn karatat). qamee-aayami manifoldsche purnapane homomorphismparyant vargikaran kele jaate.
resheey aanee topological sanrachana yaa donhi resheey topological space topology" shabdant, topological vector space) sanrachanechya antargat yetat. resheey topological jaaga hii vastavik kinwa jatil resheey jaaga aanee topological jaaga ashee asate qee resheey karye satat asatat. tyamule topological asaleli resheey jaaga hii sarvasadharanpane resheey topological jaaga nasate. 
pratyek maryadit-aayami vastavik kinwa jatil resheey jaaga hii eka resheey topological jaaga aahe, yaa arthane qee tyat eka aanee fakt ekach topology asate jyamule tee eka resheey topological jaaga banate. asha prakare, "maryadit-aayami vastavik kinwa jatil resheey jaaga" aanee "maryadit-aayami resheey topological jaaga" yaa done sanrachana samatulya aahet, mhanje paraspar antarnihit aahet. tyanusar, maryadit-aayami rekheey topological jaageche pratyek ulat karata yenyajoge rekheey parivartan heey eka homomorphism aahe. parimananchya teen kalpana (eka bijganitiy aanee done topological) maryadit-aayami vastavik resheey jagansathi sahmat aahet. tathapi, anant-aayami jaganmadhye, bhinn topology dilelya resheey sanrachaneshi susangat asoo shakatat aanee ulatya resheey parivartan samanyatah homomorphism nasatat.
sanlagn aanee prakshepak jaaga.
khalilpramane resheey jaganchya madhyamatoon sanlagn aanee prakshepak jaaga sadar karane soyiche aahe. resheey jaagechi resheey upkshetra, swatah eka resheey jaaga asalyane, eksandh nasate; tyat eka vishesh bindu, mool asato. tyala bahereel vektardvare halavalyane, ekhadyala eka sanlagn upasthan milate. teey eksandh aahe. sanlagn jaaga resheey jaaget samavisht karanyachi aavashyakta nahee, parantu tee resheey jagechya sanlagn upasthanasathi samaroop aahe. dilelya kshetravaril serve sanlagn jaaga paraspar samaroop asatat. johnn bezchya shabdat, "eka sanlagn jaaga hii eka vahak jaaga aahe jee tyache mool visarali aahe". visheshtah, pratyek resheey jaaga dekhil eka sanlagn jaaga asate.
a" yaa resheey jaaget "a" yaa sanlagn upasthanala dilelya, "a" yaa saral reshechi vyakhya, "a" yaa moodatoon olakhalya samatalasah, jeey "a" laa samantar nasel, asey binduvar "l" yaa resheey upasthanasah "a" chey chhedanbindu mhanoon kelee jaau shakate. adhik samanyatah, 'aaa' chey sanlagn upkshetra heey 'aaa' chey chhedanbindu aanee 'aaa' laa binduvar 'ell' chya resheey upkshetrasaha asate.
sanlagn upkshetra 'aaa' chaa pratyek bindu haah 'aaa' chaa 'ell' chya resheey upkshetrasaha chhed aahe. tathapi, 'l' chee kahee upkshetre 'a' laa samantar aahet; kahee arthane, teey 'a' laa anantatevar chhedatat. resheey jagechya serve resheey upasthanancha sanch, vyakhyenusar, prakshepak jaaga aahe. aanee sanlagn upkshetra "aaa" haah yogya upasanch mhanoon prakshepak jaaget antarbhut keela jaato. tathapi, prakshepak jaaga swatach eksandh aahe. prakshepak jagetil eka saral resha ("n" + 1)-aayami resheey jagechya resheey upkshetrashi sambandhit asate. adhik samanyatah, prakshepak jaagechi prakshepak upkshetra ("aine" + 1)-aayami rekheey jagechya rekheey upkshetrashi sambandhit asate aanee prakshepak jageshi samaroop asate.
asha prakare paribhashit kelelya, sanlagn aanee prakshepak jaaga bijganitiy swarupachya asatat; tyaa konatyahi kshetravar vastavik, guntaguntichya aanee adhik samanyatah asoo shakatat.
pratyek vastavik kinwa guntaguntichi sanlagn kinwa prakshepak jaaga dekhil eka topological jaaga asate. sanlagn jaaga hii eka naun-compact manifold asate; projective space hii eka compact manifold asate. vastavik prakshepak jaaget eka saral resha vartulasathi samaroop asate, tyamule aakunchansheel jagechya reshet asalelya saral reshechya ulat, tee sankshipt asate.
metrik aanee eksaman jaaga.
bindunmadhil antar metrik spacemadhye paribhashit kele jaate. metrik spacemadhil samarupatela samamiti mhanatat. pratyek metrik jaaga dekhil eka topological jaaga aahe. jarr tee metrik jagekhali asel tarr topological jages metrizable mhanatat. serve manifolds metrizable aahet.
metrik jaaget, aapan paribhashit karoo shakato 
baddh sanch aanee kachee anukram. serve kachee anukram ekatra aalyas metrik spacela purn mhanatat. pratyek apoorn jaaga samamitiyadrushtya, eka ghan upasanch mhanoon, sampoorn jaaget (purnatva) antarbhut kelee jaate. pratyek sankshipt metrik jaaga purn aahe; vastavik resha sankshipt nahee parantu purn aahe; ughade antar (0,1) apoorn aahe. 
pratyek euclidean jaaga dekhil eka sampoorn metrik jaaga asate. shivay, euclidean jaaget antarbhut asalelya serve bhaumitik kalpana tyachya metrikchya drushtine darshavalya jaau shakatat. udaharanarth, "aaa" aanee "see" yaa done dilelya bindunna bindunna saral vibhagat serve bindu "buub" asatat, jase qee "aaa" aanee "see" madheel antar heey "aaa" aanee "bii" madheel aanee "bii" aanee "see" madheel done antarachya berijaitke asate.
hasdorf pariman (dilelya sanchala sanchala lahan chendunchya sankhyeshi sambandhit) metrik jaganna lagoo hote aanee teey purnank nasalele asoo shakate (visheshtah fractalsasathi). euclidean jagesathi, housedorfche pariman "aine" chya barobariche asate.
eksaman jaaga antar parichay dett naheet, parantu tarihi ekhadyala eksaman satatya, kachee anukram (kinwa filter kinwa jaali), purnata aanee purnata vaparanyachi parvanagi detat. pratyek eksaman jaaga hii dekhil eka topological jaaga asate. pratyek "rekheey" topological jaaga (mapanaksham kinwa nahee) dekhil eksaman jaaga asate aanee tee maryadit parimanat purn asate parantu samanyatah anant parimanat apoorn asate. adhik samanyatah, pratyek parivartaneey topological gut dekhil eksaman jaaga asato. tathapi, gair-vinimayatmak topological gatamadhye done eksaman rachana asatat, eka davikadil aanee dusra ujavyakadil.
manak, kell, aateel utpadan aanee hilbert jaaga.
euclidean jaaget vahak eka resheey jaaga tayar karatat, parantu pratyek vahak sootra _ 1 chee lambi dekhil asate, kinwa shabdant, manak, sootra _ 2. aadarshasah sampann vastavik kinwa jatil resheey jaaga hii eka manak jaaga aahe. pratyek manak jaaga hii eka resheey topological jaaga aanee eka metrik jaaga donhi aahe. banach space hii eka sampoorn manak jaaga aahe. anukram kinwa karyanchya anek jaaga anant-aayami banach jaaga asatat.
ekapeksha qamee praman asalelya serve vectorchya sanchala manak jaagecha ekak chendu mhanatat. haah eka bahirmukh, madhyavarti samamiteey sanch aahe, samanyatah lambvartulakar nahee; udaharanarth, too bahubhuj (pratalamadhye) kinwa adhik samanyatah, bahupadi (aniyantrit maryadit parimanat) asoo shakato. samantar chaturbhuj niyam (jyala samantar chaturbhuj olakh dekhil mhanatat) 
samanyatah samanya jaganmadhye ayashasvi hote, parantu euclidean jaganmadhil vectorsathi teey lagoo hote, jeey vectorche varg euclidean manak heey swatahsah tyache antargat utpadan aahe, sootra _ 4 yaa vastusthitivarun yete.
aateel utpadan jaaga hii eka vastavik kinwa jatil resheey jaaga aahe, jee dvirupi kinwa anukrame sesquilinear swarupat sampann aahe, jee kahee atinche samadhan karate aanee tyala aateel utpadan mhanatat. pratyek aateel utpadan jaaga dekhil eka samanya jaaga aahe. jarr aanee keval jarr too samantar aakruticha niyam purn karato, kinwa tyachya samatulya, jarr tyacha ekak chendu lambvartulakar asel, tarr eka manak jaaga aateel utpadan jagekhali asate. vectormadhil coone aateel utpadan jaganmadhye paribhashit kele jaatat. hilbert jaagechi vyakhya sampoorn antargat utpadan jaaga mhanoon kelee jaate. (kahee lekhak aagrah dharatat qee teey guntaguntiche asale pahije, tarr itar lock vastavik hilbert jaaga dekhil swikartat.) anukram kinwa karyanchya anek jaaga anant-aayami hilbert jaaga asatat. quantum siddhantasathi hilbert space khoop mahattvachya aahet.
serve vastavik aateel utpadan jaaga paraspar samaroop aahet. euclidean jaaga hii vastavik antargat utpadan jaaga aahe jee tyache mool visarali aahe asey konihi mhanoo shakate.
gulgulit aanee remanian manifolds.
gulgulit manifoldsna 'space' mhatale jaat nahee, parantu asoo shakate. pratyek gulgulit manifold heey topological manifold asate aanee teey maryadit-aayami resheey jaaget antarbhut kele jaau shakate. maryadit-aayami resheey jaaget gulgulit prushthbhag gulgulit anekpat asatatah udaharanarth, lambvartulakar prushthbhag gulgulit anekpat asato, polytope nasato. vastavik kinwa guntaguntichya maryadit-aayami resheey, sanlagn aanee prakshepak jaaga dekhil gulgulit anek putt asatat.
tyachya pratyek binduvar, gulgulit manifoldmadhil gulgulit margavar eka sparshika vahak asato joou yaa binduvar manifoldchya sparshika jageshi sambandhit asato. gulgulit anek patanparyant sparshika jaaga yaa resheey jaaga asatat. gulgulit manifoldvaril gulgulit karyacha farak pratyek binduvaril sparshika jagevar eka resheey karya pradan karato. 
remanion manifold kinwa remain space haah eka gulgulit manifold aahe jyachya sparshika jaaga kahee atinche samadhan paribhashit aateel utpadananni sampann aahet. euclidean jaaga dekhil reeman jaaga aahet. euclidean jaganmadhil gulgulit prushthbhag heey reeman jaaga asatat. hyperbolic space hii dekhil eka reeman space aahe. reeman jaaget vakrareshechi lambi asate aanee done bindunmadhil sarvat lahan vakrareshechi lambi antar asha prakare paribhashit karate qee reeman jaaga hii metrik jaaga asate. eka binduvar binduvar done vakranmadhil coone haah tyanchya sparsh reshanmadhil coone asato.
sparshikechya jaganvaril aateel utpadananchi sakaratmakata maaf karoon, samanya sapekshatesathi khoop mahattvachya asalelya laurentzian jagansah, chhadm-riman jaaga midavata yetat.
mojata yenyajogi, mojata yenyajogi aanee sambhavyatechi jaaga.
(bhaumitik vastunche) khand rakhoon thevat antar aanee coone maaf karoon mojamapachya siddhantaparyant pohochata yete. parimanavyatirikt, sambhavyata siddhantachya andreau kolmogorovchya drushtikonanusar, eka mojmap kshetrafal, lambi, vastuman (kinwa charge) vitaran aanee sambhavyata vitaran yaa kalpananche samanyikaran karate.
shastriy ganitacha "bhaumitik bhag" haah keval bindunchya sanchapeksha adhik niyamit asato. sharirachi seema shoonya aakarmanachi asate. asha prakare, sharirache aakarman heey tyachya aateel bhagache aakarman asate aanee aateel bhag ghananchya anant kramane sampato. yaulat, bindunchya aniyantrit sanchachi seema shunyetar aakarmanachi asoo shakate (udaharaneh dilelya ghanatil serve tarksangat binduncha sanch). mojmap siddhantane aakarmanachi kalpana, tathakathit mojata yenyajogya sanchanchya vishal vargaparyant vistaranyat yash midavale. kharey tarr, mapan naan karata yenyajoge sanch anuprayoganmadhye jawaljawal kadhich hott naheet.
vyakhyenusar mojata yenyajogya jaaget dilele mojata yenyajogya sanch, mojata yenyajogya karye aanee nakashe darshavtat. topological jaageche mojata yenyajogya jaaget rupantar karnyasathi, ekhadi vyakti tyala borel sanchansah pradan karate, haah sarvat lokpriy paryay aahe, parantu ekmev paryay nahee. (bair sanch, sarvatrikpane mojata yenyajoge sanch ii. dekhil kadhikadhi vaparale jaatat.) 
topology heey boreldware advitiyapane nirdharit kele jaat nahee, udaharanarth, samanya topology aanee vegale karata yenyajogya hilbert jagevaril kamkuvat topology tyach borelala karanibhoot tharate.
pratyekjan kahee topologycha borel nasato.
pratyakshat, konatyahi topologychi parva naan karata sanchanchya (kinwa karyanchya) dilelya sangrahadvare aii tayar kele jaau shakate. mojata yenyajogya jaagecha pratyek upasanch swatach mojata yenyajogi jaaga aahe.
manak mojata yenyajogya jaaga (jyanna manak borel jaaga dekhil mhanatat) visheshtah compact jaganshi kahee samya asalyamule upayukt aahet (eom pahaa). manak mojata yenyajogya jaganmadhil pratyek bijective mojata yenyajoge mapping heey eka samarupata aahe; mhanje, ulat mapping dekhil mojata yenyajoge aahe. aanee asha jaganmadhil nakasha mojata yeil jarr aanee keval jarr tyacha aalekh utpadan jaaget mojata yeil. tyachpramane, compact metrik spacemadhil pratyek bijective nirantar mapping heey eka homomorphism aahe; mhanje, inverse mapping dekhil nirantar aahe. aanee jarr aanee keval jarr tyacha aalekh utpadanachya jaaget bund asel tarach asha jaganmadhil nakasha tayar karane satat chaloo rahate.
euclidean jaaget (aanee adhik samanyatah, sampoorn vegale karata yenyajogya metrik jaaget) thevalele pratyek borel, borelane sampann asalele eka pramanit mojata yenyajoge sthan aahe. serve mojata naan yenyajogya pramanit jaaga paraspar samaroop aahet.
mojmap jaaga hii mojamapane sampann asaleli mojata yenyajogi jaaga aahe. lebsgue mojmap asaleli euclidean jaaga hii mojmap jaaga aahe. ektrikaran siddhant mojmap jaaget ekatmata aanee mojata yenyajogya karyanche ektrikaran paribhashit karato.
shoonya sanch mhanoon olakhalya olakhalya 0 mojamapache sanch nagany aahet. tyanusar, "modd 0 samarupata" hii purn mojamapachya upasanchanmadhil samarupata mhanoon paribhashit kelee jaate (mhanje, nagany purkasah).
sambhavyata jaaga hii eka mojmap jaaga aahe, jyat sampoorn jaageche mojmap 1 chya barobariche asate. sambhavyata jaganchya konatyahi kutumbache (maryadit kinwa nahee) gunakar hii sambhavyata jaaga asate. yaulat, sarvasadharanpane mojmap jagansathi, keval maryadit anek jaganche gunakar paribhashit kele jaate. tyanusar, anek anant-aayami sambhavyata upay aahet (visheshtah, gossian upay), parantu anant-aayami lebsg upay naheet.
manak sambhavyatechya jaaga visheshtah upayukt aahet. manak sambhavyatechya jaaget sashart apeksha hii sashart mojmapapeksha avibhajya manali jaau shakate (niyamit sashart sambhavyata, mojamapache vighatan dekhil pahaa). done manak sambhavyatechya jaaga dilyas, tyanchya mojmap algebranche pratyek samarupata kahee mojmap jatan nakashadvare prerit hote. pramanit mojata yenyajogya jagevaril pratyek sambhavyateche mojmap pramanit sambhavyatechya jagekade netat. manak sambhavyata jaganchya kramache (maryadit kinwa nahee) gunakar heey manak sambhavyata jaaga aahe. serve anoo-nasaleli manak sambhavyata jaaga paraspar samaroop modd 0 aahet; tyapaiki eka mhanje lebsg mojmapasah antar (0,1).
yaa jaaga qamee bhaumitik aahet. visheshtah, itar serve jagansathi (eka swarupat kinwa kinwa swarupat) lagoo honari pariman kalpana, mojata yenyajogi, mojata yenyajogi aanee sambhavyata jagansathi lagoo hott nahee.
gair-vinimayatmak bhoomiti.
ganitache vishleshan mhanoon olakhalya olakhalya ganitachya saiddhantik abhyasamule 20 vya shatkachya suruvatila vastavik-moolya kinwa jatil-moolya asalelya karyanchya resheey jagancha vichar keela gela. yachee sarvat junie udaharane mhanje karyasthale, pratyekane swatachya samasyanchya vargala anukulata darshavali. yaa udaharananmadhye anek samaan vaishishtye samayik kelee gelee aanee hii vaishishtye lavkarach hilbert jaaga, banach jaaga aanee adhik samanya topological vector jaganmadhye amoort karnyat aalee. ganiteey samasyanchya vistrut shrenichya nirakaranasathi hii eka shaktishali sadhansamagri hotee.
sarvat tapshilvar mahitee banach algebra navachya jaganchya vargadvare ghetali gelee. heey satat gunakar karyasah ektritpane banach space aahet. eka mahattvache prarambhik udaharan mhanje 'x' mojmap jagevaril moolatah mojata yenyajogya bandhane asalelya karyanche banach bijganit. functionscha haah sanch binduvar jode aanee scalar gunakarantargat eka banach space aahe. bindunihay gunakaramule, tee eka vishesh prakarchi banach jaaga banate, jyala aata parivartaneey wonn neuman bijganit mhanatat. binduvar gunakar haah "x" vareel varg samakalit karyanchya hilbert jaaget yaa bijganitache pratinidhitva tharvato. johnn wonn newmanche suruvatiche nirikshan asey hote qee heey patravyavahar dekhil ulat kaam karateh kahee saumya tantrik gruhitkannusar, hilbert spaceveril pratinidhitvasaha eka parivartaneey wonn neuman bijganit ektritpane mojamapachi jaaga nirdharit karate aanee hii done rachana (wonn neuman bijganit adhik pratinidhitva aanee mojamapachya jaagechi) paraspar ulat aahet.
tyanantar wonn newmanne prastavit kele qee naun-commutative wonn neuman algebrascha bhaumitik arth asava, jase commutative wonn neuman algebras karatat. francis murre yanchyasobat miloon tyane wonn neuman algebrasche vargikaran kele. thet avibhajya rachana konatyahi wonn neuman bijganitache "ghatak" navachya sopya algebranchya sangrahat kase vibhajan karave heey darshavate. wonn neuman aanee murre yanni ghatakanche teen prakaranmadhye vargikaran kele. prakar I haah parivartanatmak prakaranashi jawaljawal sarkhach hota. prakar 2 aanee 3 madhye naveen ghatana pradarshit jhalya. prakar 2 wonn neuman bijganitene eka bhoomiti yaa vishisht vaishishtyasah nirdharit kelee qee pariman keval eka purnank nasoon konatihi nakaratmak nasaleli vastavik sankhya asoo shakate. prakar III algebra heey asey hote jeey prakar I kinwa II navhate aanee anek dashakanchya prayatnannantar, heey prakar II ghatkanshi jawalun sambandhit asalyache siddha jhaale.
function spaceschya bhoomitisathi thoda vegala drushtikon tyach velee vikasit jhaala jevha phaun neuman aanee murre yanni ghatakanchya vargikaranavar kaam kele. haah drushtikon yethe siddhant aahe, prerak udaharan sootra 5 aahe, jethe "x" hii sthanik patalivar sankshipt hosdorf topological space aahe. vyakhyenusar, haah 'x' vareel satatchya jatil-mulyanchya karyancha bijganit aahe joou anantatevar nahisa hoto (jyacha sadpatal arth assa aahe qee tumhi nivadalelya bindupasoon jitke door jaata, karya shoonyachya jawal jaate). gelfand-naymark pramey soochit karato qee parivartaneey aanee bhaumitik vastunmadhye parasparsanvad aaheh pratyek parivartaneey heey kahee sthanikpane sankshipt asalelya 'x' yaa 'hosdorff' jagesathi '5' yaa sootrachya rupat aahe. parinami, sthanik patalivar sankshipt asalelya hasdorf jagancha abhyas karane shakya aahe, keval parivartaneey gair-parivartaneey bhoomitichya drushtine, jarr 'gair-parivartaneey jaaga' x 'ashee ekhadi gosht asel tarr tee gair-sanvadatmak abhyasasathi prerana mhanoon ghetali jaate, tarr tyache sootra _ 5 eka gair-sanvadatmak asel; jarr yaa astitvat nasalelya vastunvar gelfand-nemark pramey lagoo kele tarr, jaaga (parivartaneey kinwa nahee) tashach asateel, kaaran gair-parivartaneey jagechya vyakhyekade thet drushtikon naslyamule, gair-vinimayatmak jaagechi' vyakhya 'kelee jaate, anek manak bhaumitik sadhananche punaravartan kele jaau shakate aanee heey gair-sanvadatmak abhyasasathi bhaumitik-prerit tantre dete.
hii donhi udaharane aata gair-vinimayatmak bhoomiti navachya kshetrachi prakarane aahet. wonn neuman algebranchi vishisht udaharane aanee anukrame naun-commutative mojmap siddhant aanee naun-commutative topology mhanoon olakhali jaatat. gair-vinimayatmak bhoomiti mhanje keval swatachya fayadhyasathi samanyatecha pathpurava karane nahee aanee tee keval eka utsukata nahee. kahee bandhkamanmadhun gair-parivartaneey jaaga naisargikaritya, agadi apariharyapane dekhil udbhavtat. udaharanarth, patang aanee dartsdware samatalatil avadhik penrosechya jhukanyacha vichar qaraa. heey eka pramey aahe qee, asha farshimadhye, patang aanee dartscha pratyek maryadit tukada anant vela disato. parinami, maryadit bhag pahoon done penrosechya zukanyamadhye farak karnyacha konatahi marg nahee. yamule serve jhukanyachya sanchala paramparik arthane topology dene ashakya hote. asey asoonahi, penroseche jhukane gair-parivartaneey nirdharit karate aanee parinami gair-parivartaneey bhoomitichya tantradvare tyancha abhyas keela jaau shakato. aanakhi eka udaharan, aanee vibhedak bhoomitimadhye khoop swarasya asalele eka, anek patanchya parnasamuhanmadhun yete. heey anek patanche "paaney" navachya lahan-aayami up-aavartanat vibhajan karnyache marg aahet, jyapaiki pratyek sthanik patalivar jawalchya itaranchya samantar aahe. serve panancha sanch topological spacemadhye banavala jaau shakato. tathapi, atarkik paribhraman udaharan darshavite qee hii topological jaaga shastriy mapan siddhantachya tantransathi durgam asoo shakate. tathapi, foliationchya pananchya jageshi sambandhit eka gair-parivartaneey wonn neuman bijganit aahe aanee punha ekada, heey anyatha samajanyajogi jaaga eka changli bhaumitik rachana dete.
yojana.
bijganitiy bhoomiti bahupadi samikarananchya bhaumitik gunadharmancha abhyas karate. bahupadi haah eka prakarcha karya aahe joou jode aanee gunakar yaa moolbhut ankganitiy kriyanmadhun paribhashit keela jaato. yamule teey bijganitashi jawalun jodalele aahet. bijganitiy bhoomiti shuddha bijganitachya prashnanna bhaumitik tantre lagoo karnyacha marg pradan karate aanee tyaulat.
1940 chya dashakapurvi, bijganitiy bhoomiti keval jatil sankhyanvar kaam karat hotee aanee sarvat moolbhut vividhata mhanje prakshepak jaaga. prakshepak jaagechi bhoomiti drushtikonachya siddhantashi jawalun sambandhit aahe aanee tyachya bijganitache varnan eksandh bahupadidvare kele jaate. itar serve prakaranchi vyakhya prakshepak jaageche upasanch mhanoon kelee gelee. projective prakar heey eksandh bahupadichya sanchadvare paribhashit kelele upasanch hote. prakshepak vividhatechya pratyek binduvar, sanchatil serve bahupadi shoonyachya barobarichya asane aavashyak hote. resheey bahupadichya shoonya sanchache purak mhanje eka sanlagn jaaga aanee sanlagn vividhata mhanje prakshepak vividhateche sanlagn jageshi chhed.
andrey weilne pahile qee jyamiteey tark kadhikadhi sankhya-saiddhantik paristhitit lagoo keela jaau shakato jithe prashnatil jaaga swatantra kinwa agadi maryadit asoo shakatat. yaa kalpanechya pathpuravyasathi, aamhi bijganitiy bhoomiticha paya punha lihila, donhi, bijganitiy bhoomitila jatil sankhyanvaril avalambitvapasun mukt kele aanee prakshepak jaaget antarbhut nasalelya "amoort bijganitiy prakar" sadar kele. hyanna aata fakt 'jaatee' mhanatat.
sarvat aadhunik bijganitiy bhoomitichya antargat asalelya jaagecha prakar heey velchya amoort bijganitiy prakaranpeksha adhik samanya aahe. alekzandar grauthendic yanni yachee suruvat kelee hotee aanee tyala yojana mhanatat. yojana siddhantachi eka prerana ashee aahe qee bahupadi hii karyanmadhye asamanyapane sanrachit asatat aanee parinami bijganitiy prakar kathor asatat. adhogati jhalelya paristhiticha abhyas karnyacha prayatna kartana heey samasya nirman karate. udaharanarth, vartulavaril jawaljawal konatyahi jodya bindunmule secont resha navachi eka anokhi resha nirdharit hote aanee jasjase done bindu vartulbhovati firatat tastashi secont resha satat badalate. tathapi, jevha done bindu bhidatat, tevha secont resha sparshikevar rupantarit hote. sparshika resha adviteey aahe, parantu yaa sanrachanechi bhoomiti-vartulavaril ekach bindu-eka adviteey resha nirdharit karnyasathi pureshi abhivyakti nahee. asha paristhiticha abhyas karnyasathi, paristhiti bighadu lagnyasathi atirikt mahitee niyukt karnyas saksham asalelya siddhantachi aavashyakta asate.
topological space haah yojanechya aadharastambhanpaiki eka aahe. topological spacemadhye satat karye asatat, parantu satatyapurn karye hii swarasya asalelya moolbhut bijganitiy sanrachaneche pratibimb padnyasathi khoop samanya asatat. tyamule, ekhadya yojnetil dusra ghatak mhanje topological spacemadhil eka aavaran, jyala "structure sheef" mhanatat. topological jagechya pratyek khulya upasanchavar, aavaran karyancha sangrah nirdisht karate, jyala "niyamit karye" mhanatat. topological space aanee structure sheef ektritpane atty purn karnyasathi aavashyak aahet, mhanje functions bijganitiy operationsmadhun yetat.
anek patanpramanech, yojananchi vyakhya asha jaaga mhanoon kelee jaate jyaa sthanik patalivar parichit jaaget tayar kelya jaatat. manifoldschya babatit, parichit jaaga mhanje euclidean jaaga. ekhadya yojnesathi, sthanik modelsna affine yojana mhanatat. affine yojana bijganitiy bhoomiti aanee parivartaneey bijganit yanchyat thet sambandh pradan karatat. parivartaneey bijganitetil abhyasachya moolbhut vastoo parivartaneey ring aahet. jarr sootra _ 8 heey eka parivartaneey ring asel, tarr sambandhit affine yojana sootra _ 9 aahe jeey sootra _ 8 chya bijganitiy sanrachaneche bhoomitimadhye bhashantar karate. yaulat, pratyek sanlagn yojana eka parivartaneey ring nishchit karate, mhanje, tyachya sanrachanechya aavaranatil jaagtik vibhaganchi ring. hii done karye paraspar viruddha aahet, mhanoon affine yojana eka naveen bhasha pradan karatat jyadvare parivartaneey bijganitetil prashnancha abhyas keela jaato. vyakhyenusar, yojnetil pratyek binducha eka khula parisar asato joou eka sanlagn yojana aahe.
asha anek yojana aahet jyaa sanlagn naheet. visheshtah, prakshepak jaaga yogyatechi aut purn karatat jee sankshiptateshi samaan aahe. sanlagn yojana yogya asoo shakat naheet (yojanecha keval ekach bindu asato asha kshullak paristhiti vagalata) aanee mhanoon konatihi prakshepak jaaga hii sanlagn yojana nahee (shoonya-aayami prakshepak jaaga vagalata). prakshepak yojana, mhanje prakshepak jaageche bund sadasya mhanoon mhanoon yojana heey yojananche ekmev sarvat mahattvache kutumb aahe.
yojananche anek samanyikaran suru karnyat aale aahe. maykal artinane bijganitiy jaagechi vyakhya, étale rupankan paribhashit paribhashit samatulya sambandhandware yojanecha bhag mhanoon kelee. bijganitiy jaaga ekach velee adhik lavchik asatana yojananche anek upayukt gunadharm rakhoon thevatat. udaharanarth, anek modulischya jaaga bijganitiy jaaga aahet heey dakhavanyasathi keel-morrie pramey vaparale jaau shakate.
bijganitiy jagepeksha adhik samanya mhanje eka deligney-mufford stack. dii. emm. stack heey yojnansarkhech asatat, parantu teey ekatmatela parvanagi detat jyanche varnan keval bahupadichya drushtine kele jaau shakat nahee. orbifolds anek patansathi karatat asha yojanansathi teey samaan bhoomika bajavtat. udaharanarth, mool bhovati firanyachya maryadit gatadware sanlagn pratalacha bhag eka vishisht-mumford sanch nirman karato joou eka yojana kinwa bijganitiy jaaga nahee. moodapasoon door, samooh krutidvare bhagakar vartulavar samaan antarachya bindunche maryadit sanch olakhato. parantu mool thikani, vartulat keval ekach bindu asato, mool swatach aanee gut krutee haah bindu nishchit karate. bhagank dm stackmadhye, tathapi, haah bindu bhagank asanyachya atirikt detasah yeto. yaa prakarchi parishkrut rachana moduli spaceschya siddhantat upayukt aahe aanee kharey tarr, tee moolatah bijganitiy vakranchya moduliche varnan karnyasathi sadar karnyat aalee hotee.
aanakhi samanyikaran mhanje bijganitiy stack, jyanna artin stack dekhil mhanatat. dii. emm. stack heey maryadit gut krutindware bhaganpurte maryadit asatat. moduli siddhantatil anek samasyansathi heey purese asale taree, itaransathi teey khoop pratibandhatmak aahe aanee artin stack adhik samanya bhaganna parvanagi detat.
topo.
well anumananvaril grothendicchya karyat, tyane aata grauthendic topology navachya topologycha eka naveen prakar sadar keela. topological space (samanya arthane) "nikatata" yaa kalpanela svayamsiddha karate, done bindu ekach khulya sanchatil anek thikani asateel tarach teey jawal asatat. yaulat, grauthendic topology svayamsiddhatemule 'aachchadan' hii kalpana tayar hote. jaageche aavaran mhanje upasthanancha sangrah jyamadhye sanyuktapane sabhovatalachya jaagechi serve mahitee asate. aavarananchya sandarbhat aavarananchi vyakhya kelee jaat asalyane, grauthendic topologykade aavarananchya siddhantache svayampreritkaran mhanoon dekhil pahile jaau shakate.
grothendicchya tyachya topologivaril karyamule too topoichya siddhantakade gela. recollets ett semiles' yaa tyanchya aatmacharitrat tyanni tyanna 'sarvat vishal sankalpana' mhatale aahe. sthanik mahitee vyakt karnyasathi aavaran (ektar topological jaaget kinwa grauthendic topologychya sandarbhat) vaparale jaate. serve shivhchya shrenimadhye sthanik mahitee vyakt karnyache serve sambhavya marg asatat. topological jaaga bindunpasoon tayar kelya jaat asalyane, jyaa swatah eka prakarchi sthanik mahitee aahet, tyamule shivhachi shreni mool jaagechi jaaga mhanoon vaparali jaau shakate. grothendicne parinami topochi vyakhya shivhachi shreni mhanoon kelee aanee topoicha abhyas tyanchya swatachya hakkat swarasya asalelya vastoo mhanoon keela. hyanna aata grauthendic topoi mhanatat.
pratyek topological jaaga topo nirdharit karate aanee tyaulat. ashee topological jaaga aahet jithe sambandhit topose ghetalyane mahitee gamavate, parantu yaa samanyatah pathological manalya jaatat. (eka aavashyak aanee pureshi aut mhanje topological jaaga eka shant jaaga asavi.) yaulat, asey topoi aahet jyancha sambandhit topological space mool topo capture karat naheet. parantu, pathological asanyaivaji, heey topoi ganitiyadrushtya khoop swarasyapoorn asoo shakatat. udaharanarth, grothendiccha etel cohomologycha siddhant (jyamule akheris well anumanancha purava midala) yojanechya etel topomadhye cohomology mhanoon shabdbaddh keela jaau shakato aanee heey topo topological spacemadhun yet naheet.
topological jaaga pratyakshat locals navachya atishay vishesh topoikade netat. topological jagechya khulya upasanchancha sanch jaali nishchit karato. topological jagesathichya svayamsiddhatemule yaa jaali sampoorn uchchabhru algebra asatat. sthanancha siddhant yala tyacha prarambhik bindu mhanoon gheto. sthanikanchi vyakhya sampoorn uchchank bijganit mhanoon kelee jaate aanee topological spacesche prathamik gunadharm yaa atinmadhye punha vyakt kele jaatat aanee tyanna nakarle jaate. sthanikanchi sankalpana topological jagepeksha adhik samanya asalyache disun yete, jyamadhye pratyek shant topological jaaga eka adviteey sthan nirdharit karate, parantu anek manoranjak thikane topological jaganmadhun yet naheet. kaaran sthanikanna guna asanyachi garaj nasate, sthanikanchya abhyasala kahee pramanat vinodane nirathak topology mhanatat.
topoi ganiteey tarkshastrashi sakhol sambandh dekhil darshavite. pratyek grauthendic topomadhye eka vishesh aavaran asate jyala subobject classifier mhanatat. heey up-object vargikaran serve sambhavya satya mulyanchya sanchapramane karya karate. sanchanchya toponmadhye, up-object vargikarankarta heey 'khote' aanee 'kharey' shhee sambandhit sanch sootra _ 11 aahe. parantu itar topoimadhye, subobject vargikaran adhik klisht asoo shakate. lauvere aanee tearney yanni olakhale qee up-object vargikarankartyachya svayamsiddhatemule adhik samanya prakarche topo midale, jeey aata prathamik topo mhanoon olakhale jaatat aanee teey prathamik topoi antardnyani tarkshastrache model hote. tarkshastrapasun teey bhoomitiparyant sadhane lagoo karnyacha eka shaktishali marg pradan karnyavyatirikt, yamule tarkshastrat bhaumitik paddhatincha vapar karane shakya jhaale.
jaaga aanee rachana.
kevin carlsonchya matey,
tathapi, "rachana" chee eka samanya vyakhya borbakine prastavit kelee hotee; tyat varr namood kelelya serve prakarchya jaaga samavisht aahet, (jawaljawal?) aataparyant vaparlya jaanarya serve prakarchya ganiteey sanrachana aanee barech kahee. heey samarupatechi samanya vyakhya pradan karate aanee samaroop sanrachananmadhil gunadharmanchya hastantaranache samarthan karate. tathapi, ganiteey saravat tyacha sakriyapane kadhihi vapar keela gela nahee (swatah borbaki yanni lihilelya ganiteey granthanmadhyehi nahee). robbert reed yanni lio corey yanchya pustakachya punaravalokanatil shevatche vakye yethe aaheth
ganiteey sanrachananvishayi adhik mahitisathi wikipediah ganiteey rachana, ganiteey sanrachanechya samatulya vyakhya aanee sanrachanechi vahtook pahaa.
bhaumitik 'rikt jaaga' aanee bijganitiy 'sanrachana' yanchyatil farak kadhikadhi spasht asato, tarr kadhikadhi durmil asato. spashtapane, gut bijganitiy aahet, tarr euclidean jaaga bhaumitik aahet. ringsvaril modules heey gatanpramanech bijganitiy asatat. visheshtah, jevha ring kshetra asalyache disate, tevha module eka resheey jaaga asalyache disate; teey bijganitiy aahe qee bhaumitik? visheshtah, jevha teey maryadit-aayami asate, vastavik sankhyanpeksha jaast asate aanee antargat utpadanane sampann asate, tevha teey euclidean space banate; aata bhaumitik. (bijganitiy?) vastavik sankhyanche kshetra (bhaumitik?) kharee resha. tyache bijganitiy samapan, (bijganitiy?) jatil sankhyanche kshetra, (bhaumitik?) jatil viman. heey sarvapratham (bijganit kinwa bhoomitiaivaji) "aapan vishleshan karato teey thikan" aahe.
gair-vinimayatmak bhoomiti", "yojana" aanee "topoi" upavibhag vagalata vareel "prakaranchya jaganmadhye" upcharit keleli pratyek jaaga, kahee atirikt sanrachanesah sampann asalela sanch (burbakinusar sanrachanecha "mukhya aadhar sanch") aahe; aadhar sanchatil ghatakanna sahasa yaa jaageche "bindu" mhanatat. yaulat, bijganitiy sanrachanechya (aadhar sanch) ghatkansathi samanyatah 'bindu' mhatale jaat nahee.
tathapi, kahivela eka vyakti ekapeksha jaast mukhya aadhar sanch vaparte. udaharanarth, dvi-aayami prakshepak bhoomiti done aadhar sanch, binduncha sanch aanee reshancha sanch yadvare aupcharik kelee jaau shakate. shivay, prakshepak pratalanche eka lakshavedhi vaishishtya mhanje bindu aanee reshanni kelelya bhoomikanchi samamiti. qamee bhaumitik udaharaneh aalekh done aadhar sanch, shirobindu sanch (jyala nodes kinwa bindu dekhil mhanatat) aanee kadaa sanch (jyala kamani kinwa resha dekhil mhanatat) yadvare aupcharik keela jaau shakato. samanyatah, maryaditpane anek pramukh aadhar sanch aanee maryaditpane anek sahayak aadhar sanch borbakidware nirdharit kele jaatat.
gair-vinimayatmak bhoomiti", "yojana" aanee vareel "topoi" upavibhaganmadhye upchar kelelya bhaumitik swadachya anek ganiteey sanrachana binduncha aadhar sanch nishchit karat naheet. udaharanarth, "arthaheen topology" topology" shabdant, bindu-mukt topology kinwa sthanik siddhant) ekach aadhar sanchane suru hote jyache ghatak topological spacemadhye khulya sanchanche anukaran karatat (parantu teey bindunche sanch naheet); keval topology aanee bindu-mukt bhoomiti dekhil pahaa.

