
uttal sett

jyaamiti mein, ack euclidean sthaan kaa ack upasamuchchay, yaa adhik aam taur parr vaastavik parr ack affine sthaan, uttal haye yadi, upasamuchchay mein kissy bhee dou binduon koo dekhate hue, upasamuchchay mein poori rekha khand hotaa haye joe unhein jodata haye. samatulya roop seey, ack uttal samooh yaa ack uttal kshetra ack upasamuchchay haye joe pratyek rekha koo ack ekal rekha khand (sambhavatah khaali) mein kaatata haye.
udaaharan key liye, ack thhos ghana ack uttal samooh haye, lekin kuchh bhee joe khokhla haye yaa ack indent haye, udaaharan key liye, ack ardhchandra aakar, uttal naheen haye.
uttal samuchchay kii seema hamesha uttal vakr hoti haye. sabhi uttal samuchchayon key pratichhedan jisamen euclidean sthaan kaa ack diyaa gayaa upasamuchchay hotaa haye, koo uttal patwar kahaa jaataa haye. yeh sabse chhota uttal samooh haye.
ack uttal falan ack vaastavik-moolya vaalaa falan haye jise ack antaral parr iss guna key saath paribhaashit kiya gayaa haye kii iska shilalekh (falan key graph parr yaa usake oopar binduon kaa sett) ack uttal samooh haye. uttal nyoonatamkaran anukoolan kaa ack upakshetra haye joe uttal samuchchay parr uttal karyon koo cumm karane kii samasya kaa adhyayan kartaa haye. uttal samuchchay aur uttal karyon key gunon key adhyayan key liye samarpit ganit kii shakha koo uttal vishleshan kahaa jaataa haye.
uttal samuchchay kii dhaarana koo neeche varnit saamaanyikrit kiya jaa sakata haye.
paribhaashaaen.
maan leejiye kii vector space yaa vaastavik sankhyaon parr ack affine space, yaa, adhik aam taur parr, kuchh krambaddh kshetra parr (isamen euclidean space shamil hain, joe affine space hain). ack upasamuchchay uttal haye yadi, sabhi key liye aur ander, line khand koo jodane aur shamil kiya jaataa haye. 
iska matlab haye kii affine sanyojan antaral mein aur sabhi key liye haye. iska tatparya haye kii affine parivartanon key tahat uttal aparivartaniya haye. isake alawa, iska tatparya haye kii ack vaastavik yaa jatil sthalvaigyaanik sadish sthaan mein ack uttal sett path-judaa huaa haye (aur isaliye bhee judaa huaa haye).
ack sett taba hotaa haye jaba line khand parr harr bindu judataa haye aur antim binduon key alawa anya bindu key sheershiya aantarik bhaag key ander hotaa haye. ack bund uttal upasamuchchay sakhti seey uttal haye yadi aur kewal tabhi jaba isake pratyek seema bindu ack charam bindu hoe.
ack samooh bilkul uttal haye yadi yeh uttal aur santulit haye.
udaaharan.
(vaastavik sankhyaon kaa samooh) key uttal upasamuchchay antaral aur bindu hain. euclidean tala key uttal upasamuchchay key kuchh udaaharan thhos niyamit bahubhuj, thhos trikon aur thhos trikon key pratichhedan hain. euclidean tri-ayami sthaan key uttal upasamuchchay key kuchh udaaharan archimedian thhos aur platonic thhos hain. capler-pointsot polyheadra gair-convex sett key udaaharan hain.
gair-samvahan sett.
ack samooh joe uttal naheen haye, usey "gair-uttal samooh" kahaa jaataa haye. ack bahubhuj joe uttal bahubhuj naheen haye, usey kabhi-kabhi avtal bahubhuj kahaa jaataa haye, aur kuchh srot aam taur parr ack gair-uttal samooh kaa arth "avtal samooh" shabd kaa upayog karate hain, lekin adhikansh adhikaari iss upayog koo pratibandhit karate hain.
ack uttal samuchchay key poorak, jaise kii ack avtal falan key shilalekh koo kabhi-kabhi "riverse uttal samuchchay" kahaa jaataa haye, vishesh roop seey ganitiya anukoolan key sandarbh mein.
guna.
uttal samuchchay mein diye gaye bindu, aur 
gair-nakaratmak sankhyaaen jaise kii, affine sanyojan 
sootra _ 1
seey sambandhit haye. jaisa kii uttal samuchchay kii paribhasha mamla haye, yeh guna uttal samuchchay koo darshaata haye.
iss tarah key ack affine sanyojan koo uttal sanyojan kahaa jaataa haye.
chauraha aur sangh.
ack sadish sthaan, ack sanlagn sthaan, yaa ack euclidean sthaan key uttal upasamuchchay key sangrah mein nimnalikhit guna hainh
bund uttal sett.
bund uttal samuchchay uttal samuchchay hotey hain jinamen unake sabhi seema bindu hotey hain. unhein "bund aadhe sthaanon" (antariksh mein bindu key sett joe ack hyperplane key ack taraf aur ack taraf sthit hain) key pratichhedan key roop mein varnit kiya jaa sakata haye.
abhee joe kahaa gayaa haye, usase yeh spasht haye kii aise pratichhedan uttal hain, aur way bund sett bhee honge. ulat saabit karane key liye, i.ee.pratyek bund uttal samuchchay koo iss tarah key pratichhedan key roop mein darshaaya jaa sakata haye, kissy koo iss roop mein sahayak hyperplane pramey kii aavashyakta hoti haye kii ack diye gaye bund uttal samuchchay aur isake baahar key bindu key liye, ack bund aadhaa-sthaan hotaa haye jisamen shamil hotaa haye aur naheen. sahayak atee-samtal pramey karyatmak vishleshan key hunn-banach pramey kaa ack vishesh mamla haye.
uttal samuchchay aur aayat.
samtal mein ack uttal shareer (ack uttal samooh jiska aantarik bhaag khaali naheen haye) honah chaahiye. humm ack aayat "r" koo iss tarah seey likh sakate hain kii "r" kii ack samroop prati "r" key baare mein pariveshtit hoe. sakaratmak samroopata anupaat adhikatam 2 haye aurah
sootra _ 2
blaske-santalo aarekh.
sabhi samtal uttal nikaayon key nirdhaarit sootra _ 3 koo uttal shareer vyaas "d", isake inradius "r" (uttal shareer mein nihit sabse badaa vritt) aur isake paridhi "r" (uttal shareer vaalaa sabse chhota vritt) key sandarbh mein maapdandit kiya jaa sakata haye. vaastav mein, iss samooh koo asamaantaon key samooh dwara varnit kiya jaa sakata haye
sootra _ 4
sootra _ 5
sootra _ 6
sootra _ 7
aur ise falan "jee" kii chhavi key roop mein dekhaa jaa sakata haye joe ack uttal shareer koo ("aar"/"aar", "dee"/2 "aar") dwara diye gaye bindu taka maanchitrit kartaa haye. iss falan kii chhavi koo a ("r", "d", "r") blatchke-santalo aarekh key roop mein jaanaa jaataa haye.
vaikalpik roop seey, sett sootra _ 3 koo isaki chaudaai (kissy bhee dou alag-alag samanantar samarthan hyperplane key beach sabse chhoti doori), paridhi aur kshetra dwara bhee parametrise kiya jaa sakata haye.
anya sampattiyan.
maanaa "x" ack topological vector space haye aur sootra _ 9 uttal haye. 
uttal patwar aur minkowski yog.
uttal patwar.
vector space kaa pratyek upasamuchchay ack sabse chhote uttal samooh (jise uttal patwar kahaa jaataa haye) key bheetar nihit hotaa haye, arthaat sabhi uttal seton kaa pratichhedan jisamen shamil hotaa haye. uttal-hull prachaalak conve () mein patwar prachaalak key vishisht guna hotey hainh
uttal-kull sanchaalan kii aavashyakta uttal samuchchay key samooh key liye ack jaali banaane key liye hoti haye, jisamen "zwine" operation dou uttal samuchchay key sangh kaa uttal patwar hotaa haye.
sootra _ 20
uttal samuchchay key kissy bhee sangrah kaa pratichhedan swayam uttal hotaa haye, isaliye ack (vaastavik yaa jatil) sadish sthaan key uttal upasamuchchay ack poorna jaali banaate hain.
minkovsky jodd.
ack vaastavik sadish-sthaan mein, dou (gair-khaali) samuchchay kaa "minkowski yog", aur, yog-samuchchay seey sadish tatva-vaar jodd dwara gathit samuchchay key roop mein paribhaashit kiya jaataa haye.
sootra _ 21
adhik aam taur parr, (gair-khaali) samuchchay key ack seemit parivaar kaa "minkowski yog" vactors key tatva-vaar jodd dwara gathit samooh haye.
sootra _ 22
minkowski key atirikt, kewal shoonya sadish vaale "shoonya samooh" kaa vishesh mahatva haih ack sadish sthaan key pratyek gair-khaali upasamuchchay key liye
sootra _ 23
beejganitiya shabdawali mein, minkovsky jodd (gair-khaali seton key sangrah parr) kaa pehchaan tatva haye.
minkowski key uttal patwar.
minkovsky jodd uttal patwar lene key sanchaalan key sambandh mein achchi tarah seey vyavhaar kartaa haye, jaisa kii nimnalikhit prastaav dwara dikhaaya gayaa haih
ack vaastavik sadish-sthaan key upasamuchchay honey dein, unake minkowski yog kaa uttal patwar unake uttal patwar kaa minkowski yog haye.
sootra _ 24
yeh parinam aam taur parr gair-khaali seton key pratyek seemit sangrah key liye adhik hotaa haih
sootra _ 25
ganitiya shabdawali mein, minkowski yog aur uttal patwar banaane key sanchaalan awagaman sanchaalan hain.
minkovsky uttal samuchchay kaa yog.
dou compact uttal samuchchay kaa minkovsky yog compact haye. ack compact uttal sett aur ack bund uttal sett kaa yog bund haye.
1966 mein diadone dwara saabit nimnalikhit prasiddh pramey, dou bund uttal upasamuchchay key antar koo bund karane key liye paryaapt shart detaa haye. yeh ack gair-khaali uttal upasamuchchay "s" key mandee shanku kii avadharana kaa upayog kartaa haye, jise iss prakaar paribhaashit kiya gayaa haih
sootra _ 26
jahan yeh samooh ack uttal shanku haye jisamen sootra _ 27 aur santoshjanak sootra _ 28 haye. dhyaan dein kii yadi "s" bund haye aur uttal haye tou sootra _ 29 bund haye aur sabhi sootra _ 30 key liye,
sootra _ 31
pramey (diadone). maan leejiye kii "a" aur "b" ack sthaaniya roop seey uttal sthalvaigyaanik sadish sthaan key gair-khaali, bund aur uttal upasamuchchay houn jaise kii sootra _ 32 ack raikhik upasthaan haye. yadi "a" yaa "b" sthaaniya roop seey compact haye tou "a" − "b" bund haye.
uttalata key liye saamaanyikaran aur vistaar.
euclidean sthaan mein uttalata kii dhaarana koo kuchh yaa anya pahaluon mein paribhasha koo sanshodhit karke saamaanyikrit kiya jaa sakata haye. samanya naam "saamaanyikrit uttalata" kaa upayog kiya jaataa haye, kyonki parinami vastuen uttal samuchchay key kuchh gunon koo banaae rakhti hain.
taaraa-samvahan (taaraa-aakar) sett.
ack vaastavik yaa jatil sadish sthaan mein ack sett honey dein. yadi koee aisaa taramandal maujood haye jisamen kissy bhee bindu seey rekha khand nihit haye tou vaha taramandal (taramandal) haye. isaliye ack khaali bahirmukhi samooh hamesha taramandal hotaa haye lekin ack taramandal samooh hamesha bahirmukhi naheen hotaa haye.
orthogonal uttalata.
saamaanyikrit uttalata kaa ack udaaharan orthogonal uttalata haye.
euclidean sthaan mein ack sett koo orthogonally convex yaa ortho-convex kahaa jaataa haye, yadi koee bhee khand dou binduon koo jodane vaale kissy bhee nirdeshank aksh key samanantar poori tarah seey bheetar sthit haye. yeh saabit karna aasaan haye kii orthoconvex sett key kissy bhee sangrah kaa ack pratichhedan orthoconvex haye. uttal samuchchay key kuchh anya guna bhee maanya hain.
gair-euclidean jyaamiti.
uttal samuchchay aur uttal patwar kii paribhasha praakritik roop seey jyaamitiyon taka faili hui haye joe ack bhooganitiya uttal samuchchay koo paribhaashit karke euclidean naheen hain jisamen bhooganiki samooh mein kissy bhee dou binduon koo jodane vaale bhooganiki shamil hain.
crum topology.
uttalata koo poori tarah seey krambaddh samooh key liye badhaaya jaa sakata haye joe crum topology key saath sampann haye.
jaane dou. upasthaan ack uttal samooh haye yadi pratyek jodee key liye aise bindu hain jisamen antaral nihit haye. arthaat, uttal haye yadi aur kewal tabhi jaba sabhi key liye, nihit haye.
ack uttal samooh samanya roop seey judaa huaa haih ack prati-udaaharan upasthaan {1,2,3} dwara diyaa jaataa haye, joe uttal donon haye aur judaa naheen haye.
uttal sthaan.
uttalata kii dhaarana koo anya vastuon key liye saamaanyikrit kiya jaa sakata haye, yadi uttalata key kuchh gunon koo swayamsiddh key roop mein chunaa jaataa haye.
ack samooh koo dekhate hue, uttalata nimnalikhit swayamsiddhon koo santusht karane key upasamuchchay kaa ack sangrah haih
inn tatvon koo uttal samuchchay kahaa jaataa haye aur iss jodee koo uttal sthaan kahaa jaataa haye. sadharan uttalata key liye, pahale dou swayamsiddh dhaaran karate hain, aur teesara tuchchh haye.
amoort uttalata kii ack vaikalpik paribhasha key liye, joe asatat jyaamiti key liye adhik upayukt haye, antimetroids seey judee "uttal jyaamiti" dekhein.
uttal sthaan.
uttalata koo ack amoort beejganitiya sanrachana key roop mein saamaanyikrit kiya jaa sakata haih ack sthaan uttal haye yadi binduon key uttal sanyojan lena sambhav haye.

