
topos

ganit mein, ack topos (,; bahuvachan topoi yaa, yaa topos) ack aisee shreni haye joe ack topological sthaan (yaa adhik aam taur parah ack site parr) parr seton kii sheev kii shreni kii tarah vyavhaar karati haye. topoi sett kii shreni kii tarah vyavhaar kartaa haye aur sthaaniyakaran kii dhaarana rakhta haye; way bindu-sett topology kaa ack pratyaksh saamaanyikaran hain. grothendic topoi beejganitiya jyaamiti mein anuprayog khojata haye; tark mein adhik samanya praathamik topoi kaa upayog kiya jaataa haye.
joe ganitiya kshetra topoi kaa adhyayan kartaa haye, usey topos siddhant kahaa jaataa haye.
grothendic topos (jyaamiti mein topos).
1940 key dashak mein ganit mein sheeves kii shuruat key baad seey, ack pramukh vishay ack sthaan parr sheeves kaa adhyayan karke ack sthaan kaa adhyayan karna rhaa haye. iss vichaar koo alekjendar grothendic ney "topos" kii dhaarana koo pesh karke samajhaaya thaa. iss dhaarana kii mukhya upayogita ganit mein aisee sthitiyon kii prachurta mein haye jahaan topological huristics bahut prabhaavi hain, lekin ack imandar topological sthaan kii kamee haye; kabhi-kabhi huristic koo aupchaarik banaane vaale topos koo dhoondhana sambhav haye. iss karyakramatmak vichaar kaa ack mahatvapurna udaaharan ack yojana kaa etel topo haye. vibhinn ganitiya sthitiyon key "saar" koo avtaar dene key liye grothendic topos kii kshamata kaa ack aur chitran una siddhanton koo jodane key liye pull key roop mein unake upayog dwara diyaa gayaa haye, joe sambhavatah bahut alag bhaashaon mein likhe jaane key bavjood, ack samanya ganitiya samagri saajha karate hain.
samatulya paribhaashaaen.
grothendic topos ack shreni "see" haye joe nimnalikhit teen gunon mein seey kissy ack koo santusht karati haye. (jeane giraud kii ack pramey bataati haye kii neeche diye gaye sabhi guna samatulya hain.)
yahaan presh ("dee") "dee" seey sett kii shreni taka vipareet functors kii shreni koo darshaata haye; iss tarah key vipareet functor koo aksar preshif kahaa jaataa haye.
giraud key swayamsiddh.
see" shreni key liye giraud key swayamsiddh siddhant hainh
antim swayamsiddh koo sabse adhik spashteekaran kii aavashyakta haye. yadi "x" "c" kii vastu haye, tou "x" parr ack "samatulya sambandh" r "" c "mein ack maanchitra" r "" "x" "×" x "haye."
iss tarah kii "c" mein kissy bhee vastu "y" key liye, prerit maanchitra home ("y", "r") → home ("y", "x") × home ("y", "x") sett home ("y", "x") parr ack sadharan samatulya sambandh detaa haye. choonki "c" mein colimit hain, isaliye humm dou maanchitron "r" "" x "" key samatulya banaa sakate hain; ise "x"/"r" kahein. yadi canonical maanchitra haye tou samatulyata sambandh "prabhaavi" haye.
yeh ack samroopata haye.
udaaharan.
giraud kii pramey pahale seey hii udaharanon kii poori soochi key roop mein "saiton parr sheeves" detee haye. haalaanki, dhyaan dein kii koee bhee samaan site aksar naheen detee haye
samatulya topoi taka badhana. jaisa kii parichay mein sanket diyaa gayaa haye, sadharan sthalaakritik sthaanon parr sheeves kaee buniyaadi paribhashaon aur topos siddhant key parinamon koo prerit karate hain.
sett aur jee-sett kii shreni.
sett kii shreni ack mahatvapurna vishesh mamla haih yeh topos siddhant mein ack bindu kii bhoomika nibhaata haye. vaastav mein, ack samooh koo ack bindu parr ack sheef key roop mein socha jaa sakata haye kyonki ekal vastu key saath ekal shreni parr functor aur kewal pehchaan rupankan sett kii shreni mein kewal vishisht sett hain.
isi tarah, kissy bhee samooh sootra key liye ack topos sootra _ 3 haye joe sootra _ 4-sett kii shreni key barabar haye. humm ise ack vastu key saath shreni parr purvanirdhaarit kii shreni key roop mein banaate hain, lekin abb aakriti kaa samooh samooh sootra _ 4 dwara diyaa jaataa haye. kyonki kissy bhee functor koo lakshya parr ack sootra _ 4-kriya deni chaahiye, yeh sootra _ 4-sett kii shreni detaa haye. isi tarah, ack samooh sootra _ 9 key liye sootra _ 9 parr purvanirdhaarit kii shreni sootra _ 9 mein vastuon key samooh dwara anukramit seton kaa sangrah detee haye, aur sootra _ 9 mein ack vastu key swachaalan mein functor key lakshya parr ack kaarravaai hoti haye.
valayakar sthaanon seey topoi.
adhik videshi udaaharan, aur topos siddhant kaa "kaaran dee 'etre", beejganitiya jyaamiti seey aatey hain. ack topo kaa mool udaaharan ack yojana key zariski topo seey aataa haye. pratyek yojana sootra 13 key liye ack site sootra 14 (khule upasamuchchay aur samavesh dwara diye gaye rupankan dwara dee gaee vastuon kaa) haye jisaki poorvasoochana kii shreni zaarisi topos sootra 15 banaati haye. lekin ack baar rupankan key vishisht vargon parr vichaar kiye jaane key baad, isake kaee saamaanyikaran hotey hain joe gair-tuchchh ganit kii or leey jaate hain. isake alawa, topoi vishuddh roop seey algebra kii shreni parr kaary karane vaalon key roop mein yojanaon key adhyayan key liye neenv detaa haye.
ack yojana aur yahaan taka kii ack dher bhee ack etel topo, ack aff. pee. pee. aff. topo, yaa ack nisnevich topo koo jodd sakata haye. topos kaa ack aur mahatvapurna udaaharan kristaliya sthal seey haye. etel topo key mamle mein, yeye anabelian jyaamiti mein adhyayan kii moolbhoot vastuon kaa nirmaan karate hain, joe beejganitiya jyaamiti mein vastuon kaa adhyayan karate hain joe poori tarah seey unake etel maulik samooh kii sanrachana dwara nirdhaarit hotey hain.
roga.
topos siddhant, kuchh arthon mein, shaastriya bindu-sett topology kaa ack saamaanyikaran haye. isaliye kissy koo rogjanak vyavhaar key puraane aur nae udaharanon koo dekhane kii ummeed karni chaahiye. udaaharan key liye, ack gair-tuchchh topo key chhed key kaaran ack udaaharan haye jisamen koee ank naheen haye (topo kii binduon kii paribhasha key liye neeche dekhein).
jyaamitiya rupankan.
yadi sootra _ 13 aur sootra _ 17 sheersh hain, tou ack "jyaamitiya rupankan" sootra _ 18 sahayak functors kii ack jodee haye ("u" ≤ ", u" ≤) (jahaan "u" ≤: "y" "" x "" koo "u" ≤: "x" "" y "" key saath chhod diyaa gayaa haye) jaise kii "u" ≤ seemit seemaon koo sanrakshit kartaa haye. dhyaan dein kii "u" ≤ swachaalit roop seey ack sahee buggle honey key kaaran colimit koo sanrakshit kartaa haye.
freed key sahayak functor pramey dwara, ack jyaamitiya rupankan "x" "" y "dene key liye ack functor" u "≤:" y "" "x" denaa haye joe seemit seemaon aur sabhi chhote colimit koo sanrakshit kartaa haye. iss prakaar topoi key beach jyaamitiya rupankan koo sthaanon key maanchitron key anuroop key roop mein dekhaa jaa sakata haye.
yadi sootra 13 aur sootra 17 sthalshastriya sthaan hain aur sootra 21 unake beach ack nirantar maanchitra haye, tou sheeves parr pullback aur push forward sanchaalan site sootra 22 key liye sambandhit topoi key beach ack jyaamitiya rupankan pradaan karate hain.
topoi key ank.
ack topos sootra _ 13 key ack bindu koo sett key topos seey sootra _ 13 taka ack jyaamitiya rupankan key roop mein paribhaashit kiya gayaa haye.
yadi "x" ack sadharan sthaan haye aur "x" "x" kaa ack bindu haye, tou joe functor ack sheef "f" koo apane danthal "fx" parr leey jaataa haye, uska ack sahee sameep hotaa haye.
("gaganchumbi imarat sheef" functor), isaliye "x" kaa ack sadharan bindu bhee ack sheersh-saiddhaantik bindu nirdhaarit kartaa haye. inhen nirantar maanchitra "x": "1" "x" key saath pullback-push forward key roop mein banaaya jaa sakata haye.
ack sthaan sootra _ 13 key etel topos sootra _ 25 key liye, ack bindu ack vastu kaa thodaa adhik parishkrit haye. antarnihit yojana sootra kaa ack bindu sootra diyaa gayaa haye _ 13 ack bindu sootra _ 25 koo phir sootra _ 32 key ack alag kshetra vistaar sootra _ 31 dwara diyaa jaataa haye jaise kii mool bindu sootra _ 34 key madhyam seey sambaddh maanchitra sootra _ 33 kaarak. phir, kaarak maanchitra sootra _ 35 yojanaon kaa ack etel rupankan haye.
adhik sateek roop seey, way "vaishvik" bindu hain. way ack topo key sthaan jaise pahalu koo pradarshit karane key liye paryaapt naheen hain, kyonki ack gair-tuchchh topo mein koee bhee naheen hoe sakata haye. "saamaanyikrit" "bindu ack topos" "y" "(paribhasha kaa charan) seey" "x" "taka jyaamitiya rupankan hain. inmein seey paryaapt sthaan jaise pahalu koo pradarshit karane key liye hain. udaaharan key liye, yadi "x" ack jyaamitiya siddhant "t" key liye vargeekaran karane vaalaa topo "s" ["t"] haye, tou saarvbhaumik guna kahata haye kii isake bindu "t" key model hain (paribhasha key kissy bhee charan mein "y").
aavashyak jyaamitiya rupankan.
ack jyaamitiya rupankan ("u" ≤ ", u" ≤) "aavashyak" haye yadi "u" ≤ kaa ack aur baaen kinare "u" haye!, yaa samkaksh (sahayak functor pramey dwara) yadi "u" ≤ naa kewal seemit balki sabhi chhoti seemaon koo sanrakshit kartaa haye.
ringed topoi.
ack ringed topos ack jodee ("x", "r") haye, jahan "x" ack topos haye aur "r" "x" mein ack commutative ring object haye. ring vaale sthaanon key adhikansh nirmaan ring vaale topoi key liye hotey hain. x" mein "r"-module vastuon kii shreni paryaapt injection key saath ack abelian shreni haye. ack adhik upayogi abelian shreni ardh-susangat "aar"-module kii upashreni haih yeye "aar"-module hain joe ack prastuti koo sweekaar karate hain.
ringed topoi kaa ack anya mahatvapurna varg, ringed space key alawa, dilligne-mumford dheron kaa etel topoi haye.
topoi kaa samroopata siddhant.
michael artin aur bairi majur ack topos ack pro-simplisal sett (homotopy taka) key antarnihit site seey judey hue hain. (ise hoe (pro-aiss. aiss.) mein vichaar karna behtar haye; edward dekhein) saral seton kii iss vyutkram pranali kaa upayog karke koee bhee "kabhi-kabhi" shaastriya topology mein ack samroopata aparivartaniya seey jodd sakata haye joe kii topos siddhant mein aparivartaniya kii ack vyutkram pranali haye. ack yojana key etel topos seey judey pro-simplisal sett key adhyayan koo etel homotopy siddhant kahaa jaataa haye. achche mamalon mein (yadi yojana notherian aur jyaamitiya roop seey ack shakha haye), tou yeh saralata-samarthak sett pro-finite haye.
praathamik topoi (tark mein topoi).
parichay.
20veen shatabdi kii shuruat seey, ganit kii pramukh swayamsiddh neenv siddhant nirdhaarit kii gaee haye, jisamen sabhi ganitiya vastuon koo antatah sett dwara darshaaya jaataa haye (karyon sahit, joe seton key beach maanchitra banaate hain). shreni siddhant mein adhik halia kaam iss neenv koo topoi kaa upayog karke saamaanyikrit karane kii anumati detaa haye; pratyek topo poori tarah seey apane ganitiya dhaanche koo paribhaashit kartaa haye. sett kii shreni ack parichit topo banaati haye, aur iss topo key bheetar kaam karna paaramparik sett-saiddhaantik ganit kaa upayog karane key barabar haye. lekin isake bajaay koee bhee kaee vaikalpik topoi key saath kaam karna chuna sakata haye. kissy bhee topo mein pasand key swayamsiddh kaa ack maanak sutrikaran samajh mein aataa haye, aur aise topoi hain jinamen yeh amaanya haye. rachanaatmak loga binaa kissy bahishkrit madhya key kaanoon key ack topo mein kaam karane mein ruchi rakhenge. yadi kissy vishesh samooh "jee" key tahat samroopata mahatvapurna haye, tou koee bhee sabhi "jee"-seton seey yukt topo kaa upayog curr sakata haye.
ack beejganitiya siddhant koo kootbaddh karna bhee sambhav haye, jaise kii samoohon key siddhant koo, ack topo key roop mein, ack vargeekaran topo key roop mein. siddhant key vyaktigat model, i.ee. hamaare udaaharan mein samooh, phir encoding topo seey sett kii shreni key liye functors key anuroop hain joe topo sanrachana kaa sammaan karate hain.
aupchaarik paribhasha.
jaba moolbhoot kaary key liye upayog kiya jaataa haye tou ack topo koo swayamsiddh roop seey paribhaashit kiya jaayega; sett siddhant koo taba topos siddhant key ack vishesh mamle key roop mein maanaa jaataa haye. shreni siddhant key aadhar parr, ack topo kii kaee samkaksh paribhaashaaen hain. nimnalikhit mein sankshipt honey kaa guna haih
ack topo ack aisee shreni haye jisamen nimnalikhit dou guna hainh
aupchaarik roop seey, ack vastu sootra _ 13 kii ack shakti vastu sootra _ 38 key saath ack jodee sootra _ 37 haye, joe nimnalikhit arthon mein sambandhon koo vargeekrit kartaa haye. 
pahala dhyaan dein kii pratyek vastu sootra 39 key liye, ack rupankan sootra 40 ("upasamuchchay kaa ack parivaar") ack upa-vastu sootra 41 koo prerit kartaa haye. aupchaarik roop seey, ise sootra 43 key saath sootra 42 koo peechhe kheenchkar paribhaashit kiya jaataa haye. ack shakti vastu kaa saarvbhaumik guna yeh haye kii pratyek sambandh iss tarah seey utpann hotaa haye, joe sambandh sootra 44 aur rupankan sootra 40 key beach ack dwi-uddeshya patrachar detaa haye.
seemit seemaon aur shakti vastuon seey koee bhee yeh praapt curr sakata haye kii
kuchh anuprayogon mein, upa-vastu vargeekaran kii bhoomika mahatvapurna haye, jabaki shakti vastuein naheen hain. iss prakaar kuchh paribhaashaaen kyaa paribhaashit kiya gayaa haye aur kyaa praapt kiya gayaa haye, isaki bhoomikaon koo ulat detee hain.
taarkik functor.
ack "taarkik functor" sheershon key beach ack functor haye joe seemit seema aur shakti vastuon koo sanrakshit kartaa haye. taarkik functor una sanrachanaon koo sanrakshit karate hain joe sheersh parr hoti hain. vishesh roop seey, way seemit colimit, upa-vastu vargeekaran aur ghaatiya vastuon koo sanrakshit karate hain.
spashteekaran.
oopar paribhaashit kiye gaye ack topo koo ack cartasian bund shreni key roop mein samajha jaa sakata haye jisake liye kissy vastu key upa-vastu kii dhaarana mein ack praathamik yaa pratham-crum kii paribhasha haye. yeh dhaarana, ack samooh key upasamuchchay, ack samooh key upsamooh aur kissy bhee beejganitiya sanrachana key adhik aam taur parr upa-beejganit kii dhaaranaon key ack praakritik shreneegat amoort key roop mein, topos kii dhaarana seey pahale kii haye. yeh kissy bhee shreni mein paribhaashit haye, naa kii kewal topoi, dusare crum kii bhasha mein, i.ee. vyaktigat rupankan key bajaay rupankan key vargon key sandarbh mein, nimnanusar. dou monic diye gaye hain "m", "n" kramshah "y" aur "z" seey "x" taka, humm kahate hain kii "m" ≤ "n" jaba ack rupankan maujood hotaa haye "p": "y" → "z" jisake liye "np" = "m", monic parr ack poorv crum koo "x" key liye prerit kartaa haye. jaba "m" ≤ "n" aur "n" ≤ "m" humm kahate hain kii "m" aur "n" samatulya hain. x" key upa-vastuein isake liye monix key parinami samatulya varg hain.
ack sheersh mein "upa-vastu" cumm seey cumm antarnihit roop seey, ack pratham-crum kii dhaarana bunn jaati haye, jaisa kii nimnanusar haye.
jaisa kii oopar ullekh kiya gayaa haye, ack topos ack shreni "see" haye jisamen sabhi seemit seemaaen hain aur isaliye vishesh roop seey khaali seema yaa antim vastu 1. taba "x": 1 "x" key roop mein "x" ω "x" key roop mein rupankan kaa ilaaj karna swabhavik haye. rupankan "f": "x" "" y "iss prakaar pratyek tatva" x "" ± "x" koo "fx" ± "y" tatva key saath maanchitran karane vaale karyon key anuroop haye, jisamen sanrachana dwara mahsoos kiya gayaa anuprayog haye.
taba koee "x" key ack upa-vastu koo "m" key samatulya varg key roop mein paribhaashit karane key baare mein soch sakata haih "x′" "x" kii chhavi samaan haye." . "x" "±" "x′" ". pakad yeh haye kii dou yaa dou seey adhik rupankan ack hii kaary key anuroop hoe sakate hain, yaani, humm yeh naheen maan sakate kii "see" iss arth mein thhos haye kii fuctor "see" (1,-): "see" → sett wafadar haye. udaaharan key liye graph kii shreni graph aur unase judey samroopata ack topo haye jiska antim uddeshya 1 ack sheersh aur ack kinare (ack swa-loop) key saath graph haye, lekin thhos naheen haye kyonki graph "jee" key tatva 1 → "jee" kewal swa-loop seey mell khaate hain aur anya kinaaron seey naheen, naa hii swa-loop key binaa sheersh. jabaki dusare crum kii paribhasha "jee" aur "jee" key sabhi swa-loopon (unake sheershon key saath) key upa-aalekh koo "jee" key alag upa-vastuon (jaba taka kii pratyek kinara naheen haye, aur pratyek sheersh mein ack swa-loop naheen haye) banaati haye, yeh chhavi-adharit naheen haye. ise graph udaaharan aur sambandhit udaharanon key liye yonada lemma key madhyam seey sambodhit kiya jaa sakata haye jaisa kii neeche diye gaye udaaharan khand mein varnit haye, lekin yeh taba pratham-crum naheen hotaa haye. topoi ack adhik amoort, samanya aur pratham-crum samadhan pradaan kartaa haye.
jaisa kii oopar ullekh kiya gayaa haye, ack topos "see" mein ack upa-vastu vargeekaran ω hotaa haye, arthaat ack tatva "tii" ω key saath "see" kii ack vastu, "see" kaa "samanya upa-vastu", jisamen yeh guna hotaa haye kii pratyek monic "aim": "aiks" "" aiks "ack advitiya rupankan" aff ":" aiks "" "ω key saath samanya upa-vastu key pullback key roop mein utpann hotaa haye, jaisa kii chitra 1 key anusaar. abb ack monic kaa pullback ack monic haye, aur" tii "sahit sabhi tatva monic hain kyonki kissy bhee vastu seey kewal ack rupankan 1 hotaa haye, jaba seey" tii "kaa pullback" aff ":" aiks "" "key saath ack monic haye. isaliye "x" seey "x" taka key morphism key saath "t" key pullback key saath "x" key liye monix bijection mein hain. baad key rupankan ney monic koo samatulya vargon mein vibhajit kiya, pratyek rupankan "f": "x" → ω, usa varg kii visheshta rupankan, jise humm "x" key upa-vastu key roop mein lete hain yaa jise "f" dwara naamit kiya jaataa haye.
yeh sub kissy bhee topo parr lagu hotaa haye, chaahe vaha thhos hoe yaa naheen. thhos mamle mein, arthaat "see" (1,-) wafadar, udaaharan key liye sett kii shreni, sthiti karyon key parichit vyavhaar taka cumm hoe jaati haye. yahaan monic "m": "x′" "x" "seey" x "taka key injection (ack-ack kaary) hain, aur joe dee gaee chhavi key saath hain {" mx " . "x" "bul" "x′" "}" "x" "kaa upa-vastu haye joe rupankan" "f" "key anuroop haih" "x" "→ ω jisake liye" "f" "−1 (" "t" ") vaha chhavi haye. ack upa-vastu key monic mein samanya roop seey kaee domen honge, joe sabhi halanki ack dusare key saath sambandh mein honge.
sankshep mein, upa-vastu vargeekaran kii yeh pratham-crum dhaarana ack topo key liye "x" key liye samaan samatulyata sambandh koo antarnihit roop seey paribhaashit karati haye jaisa kii pahale kissy bhee shreni key liye upa-vastu kii dwiteeya-crum dhaarana dwara spasht roop seey paribhaashit kiya gayaa thaa. roopvaad key ack varg parr samatulya sambandh kii dhaarana swayam aantarik roop seey dusare crum kii haye, jise topos kii paribhasha spasht roop seey kewal upa-vastu "vargeekaran" ω kii dhaarana koo paribhaashit karke spasht roop seey darkinar curr detee haye, jisase "x" kii upa-vastu kii dhaarana koo ack antarnihit parinam key roop mein chhod diyaa jaataa haye (aur isaliye naam yogya) joe isake sambandhit rupankan "f": "x" → ω dwara visheshta haye.
aage key udaaharan aur gair-udaaharan.
pratyek grothendic topo ack praathamik topo haye, lekin iska vipareet sach naheen haye (kyonki pratyek grothendic topo poorna haye, joe ack praathamik topo seey aavashyak naheen haye).
parimit samuchchay kii shreniyaan, seemit "jee"-samuchchay (ack seemit samuchchay parr ack samooh "jee" kii kriyaaen), aur seemit graph praathamik topoi hain joe grothendic topoi naheen hain.
yadi "see" ack chhoti shreni haye, tou functor shreni sett "see" (jisamen "see" seey laker sett taka key sabhi sahparivaahak functor shamil hain, jisamen praakritik parivartan rupankan key roop mein hotey hain) ack topo haye. udaaharan key liye, dou sheershon key beach kaee nirdeshit kinaaron kii anumati dene vaale prakaar key graph kii shreni graph ack topo haye. iss tarah key graph mein dou sett hotey hain, ack aij sett aur ack vertex sett, aur una seton key beach dou kaary "s", t ", pratyek kinare" e "isake srot" s "(" e ") aur lakshya" t "(" e ") koo nirdisht karate hain. iss prakaar jee. aar. aff. functor shreni sett "see" key barabar haye, jahan "see" dou vastuon "ee" aur "vee" aur dou rupankanon "aiss" key saath shreni haye, tii ":" ee "" "v" "pratyek kinare key srot aur lakshya koo kramshah detaa haye.
yoneda lemma kaa kahana haye kii "see" opp ack poorna upashreni key roop mein sett "see" mein embed kartaa haye. graph key udaaharan mein embedding "see" opp koo sett "see" kii upashreni key roop mein darshaati haye, jisaki dou vastuein "vee" hain ack-vertex noo-aij graph key roop mein aur "ee" dou-vertex vann-aij graph (donon functor key roop mein) key roop mein, aur jinake dou gair-pehchaan morphism "vee" seey "ee" (donon praakritik parivartanon key roop mein) key dou graph homomorphism hain. v" seey ack manmana graph (functor) "g" mein praakritik parivartan "g" key sheershon kaa gathan karate hain jabaki "e" seey "g" taka key isake kinaaron kaa gathan karate hain. halanki sett "see", jise humm jee. aar. aff. key saath pehchaan sakate hain, kewal "vee" yaa "ee" "dwara thhos naheen banaaya jaataa haye, functor" yuu ": jee. aar. aff.": jee. aar. aff. "," jee ", jee", jee ") aur morphism" aich ":" jee "," aich "," jee "," aich ") key liye vastu" jee "koo sett kii jodee (jee. aar. aff.," vee "," jee ") key liye bhejata haye. yaani, graph key ack rupankan koo karyon kii ack jodee key roop mein samajha jaa sakata haye, ack sheershon kaa maanchitran aur dusare kinaaron key saath, anuprayog key saath abhee bhee sanrachana key roop mein mahsoos kiya jaataa haye lekin abb kaee prakaar key "saamaanyikrit" tatvon key saath. yeh darshaata haye kii ack thhos shreni kii paaramparik avadharana key roop mein jisaki vastuon mein ack antarnihit sett hotaa haye, usey ack vastu koo kaee antarnihit seton kii anumati dekar, yaani bahuayami honey kii anumati dekar topoi kii ack vyaapak shrinkhala koo puuraa karane key liye saamaanyikrit kiya jaa sakata haye.
bindu-sanrakshan karyon key saath nukile samuchchay kii shreni "naheen" ack topos haye, kyonki isamen shakti vastuein naheen hainh yadi sootra _ 46 nukile sett sootra _ 13 kaa shakti vastu thaa, aur sootra _ 48 nukile ekal tunn koo darshaata haye, tou kewal ack bindu-sanrakshan kaary sootra _ 49 haye, lekin sootra _ 50 mein sambandh sootra _ 13 key nukile upasamuchchay key roop mein kaee hain. abelian samoohon kii shreni bhee ack topos naheen haye, isi tarah key kaaran seh pratyek samooh samroopata koo 0 seey 0 kaa maanchitran karna chaahiye.
sandarbh.
nimnalikhit granth topos key liye aasaan-gati parichay aur shreni siddhant kii mool baatein hain. way una logon key liye upayukt honey chaahiye joe ganitiya tark aur sett siddhant koo cumm jaanate hain, yahaan taka kii gair-ganitvidon key liye bhee.
topos parr grothendic moolbhoot karyah
nimnalikhit monograph mein kuchh yaa sabhi topos siddhant kaa parichay shamil haye, lekin mukhya roop seey shuruati chhatron koo puuraa naheen karate hain. badhati kathinaaee key (kathit) crum mein soochibaddh.

