nett (ganeet)

ganitamadhye, visheshtah sarvasadharan topology aanee sambandhit shakhanmadhye, jaali kinwa mooar-smith kram heey anukramachya kalpaneche samanyikaran aahe. thodakyat, anukram heey eka karya aahe jyache kshetra naisargik sankhya aahe. yaa karyacha kodomen samanyatah kahee topological space asato.
anukramachya kalpaneche samanyikaran karanyachi prerana ashee aahe qee, topologychya sandarbhat, anukram topological spacemadhil karyambaddalchi serve mahitee purnapane encode karat naheet. visheshtah, khaleel done atty, sarvasadharanpane, topological space sootra _ 2 aanee sootra _ 3 darmyanchya nakash sootra _ 1 sathee samatulya naheet.
jaree aut 1 nehmi aut 2 chee hummy dete, parantu topological jaaga donhi pratham-mojanyayogya nastil tarr ulat asane aavashyak nahee. visheshtah, done atty metrik spesesasathi samatulya aahet. jyaa jagansathi ulat dharoon aahe tyaa anukramik jaaga aahet.
netchi sankalpana, jee pratham eae. yanni sadar kelee. etch. muur aanee herman ell. 1922 madhye smithne eka anukramachi kalpana samanya karane aahe jenekarun vareel atty (sthitee 2 madhye 'anukram' chya jaagi 'nett' neey badalalya jaatil) pratyakshat topological spaceschya serve nakashansathi samatulya asateel. visheshtah, ganana karnyayogya resheey krambaddha sanchavar paribhashit karnyaivaji, nivval aniyantrit nirdeshit sanchavar paribhashit kele jaate. heey vareel atty 1 aanee 2 yaa topological spaceschya sandarbhat dharoon thevanyachya samatulya aahet, jyachyakade eka binduchya bhovati mojata yenyajoga kinwa resheey krambaddha shejarcha aadhar asane aavashyak nahee, yaa davyasarkhech pramey karnyas anumati dete. mhanoon, anukram topological spaces darmyanchya karyambaddal pureshi mahitee encode karat nasale taree, jaale karatat, kaaran topological spacessmadhil khulya sanchanche sangrah heey vartanatil nirdeshit sanchasarkhe asatat. nett' haah shabd johnn ell. yanni tayar keela hota. kelee.
metrick spaceschya sandarbhat puresha pramanat samanya nasalelya kahee sankalpananche samanyikaran karnyasathi topologymadhye vaparlya jaanarya anek sadhananpaiki jaali eka aahe. filterchi sambandhit kalpana 1937 madhye heinry carton yanni vikasit kelee hotee.
vyakhya.
jyache domain nirdeshit sanch aahe asha konatyahi karyala 'a' mhanatat. jarr heey karya kahee sanch sootra _ 2 madhye moolye ghet asel tarr tyala spashtapane 'a' asehi sambodhale jaau shakate, a heey formula _ 12 chey karya aahe jethe sootra _ 13 haah kahee nirdeshit sanch aahe. netchya kshetrachya ghatakanna 'itts' mhanatat. 
a heey eka rikt nasalele sanch sootra aahe, jeey purvkramasah ekatra aahe, samanyatah svayamchalitapane sootra _ 15 dware darshavile jaate asey gruheet dharale jaate (anyatha soochit kele nahee tarr), jyaa gunadharmansah teey dekhil () aahe, jyacha arth assa aahe qee konatyahi sootra _ 16 sathee kahee sootra _ 17 astitvat aahe, jase qee sootra _ 18 aanee sootra _ 19 
shabdat sangayache tarr, yaa gunadharmancha arth assa aahe qee konatehi done ghatak (sootra _ 13 chey) dilyas, nehmich kahee ghatak asatat jeey tyaa doghanchya varr asatat (mhanje, tyaa pratyekapeksha mothe kinwa samaan asatat); asha prakare, nirdeshit sanch ganitiyadrushtya kathor paddhatine 'disha' yaa kalpanela samanya karatat. 
naisargik sankhyanche sootra _ 21 aanee nehamichya purnank tulana sootra _ 15 purvakram heey ektritpane nirdeshit sanchache mool udaharan tayar karatat. kharey tarr, eka jaali jyache kshetra naisargik sankhya aahe too eka kram aahe kaaran vyakhyenusar, sootra _ 2 madheel kram heey sootra _ 24 teey sootra _ 25 paryantche fakt eka karya aahe, asha prakare jaali hii anukramanchi samanyikaran aahet. mahattvache mhanje, naisargik sankhyanchya ulat, nirdeshit sanch heey ekoon aurder kinwa agadi aanshik aurder asane aavashyak aahe. 
shivay, nirdeshit sanchanna sarvat mothe ghatak aanee/kinwa kamal ghatak asanyachi parvanagi aahe, mhanunach jaale vapartana, mool (kathor nasalele) purvakram sootra _ 27 aivaji prerit kathor purvakram sootra _ 26 vapartana savadhgiri balgane aavashyak aahe; visheshtah, jarr nirdeshit sanch sootra _ 28 madhye sarvat mothe ghatak sootra _ 29 asel tarr konatehi sootra _ 30 astitvat aahe, jase qee sootra _ 31 (yaulat, sootra _ 33 asey kahee sootra _ 30 astitvat aahe).
jaali varanvar anukramansah vaparlya jaanarya sanketankapramane (aanee tyatoon prerit) asalelya sanketank vaparoon darshavili jaatat. 
sootra _ 2 madheel nivval sootra _ 35 dware darshavile jaau shakate, jethe anyatha vichar karnyache kaaran nasel tarr, heey aapoaap gruheet dharale pahije qee sanch sootra _ 13 nirdeshit aahe aanee tyacha sambandhit purvakram sootra _ 37 dware darshavila gela aahe. 
tathapi, kahee lekhak, udaharanarth, kansanaivaji koneey cons sootra _ 38 vaparoon jaalisathiche sanket badaltat. 
sootra _ 2 madheel nivval sootra _ 40 mhanoon dekhil lihile jaau shakate jeey heey satya vyakt karate qee heey nivval sootra _ 41 heey eka karya sootra _ 42 aahe jyache moolya tyachya domenmadhil ghatak sootra _ 43 varr nehamichya cons sanketan sootra _ 45 aivaji sootra _ 44 dware darshavile jaate jeey samanyatah karyansah vaparale jaate (heey sadasyata sanketan anukramatoon ghetale jaat aahe). bijganitiy topologychya kshetrapramane, bharaleli disk kinwa 'bullet' heey jaalivaril yuktivad (mhanje jaalichya kshetrache ghatak sootra _ 29) jethe thewale jaatat teey sthan darshavate; heey jaali heey eka karya aahe yavar jore denyas madat karate aanee nantar tyacha sandarbh detana lihave lagnare nirdeshank aanee itar chinhanchi sankhya dekhil qamee karate.
jaali pramukhyane vishleshan aanee topologychya kshetrat vaparali jaatat, jithe tyancha vapar anek mahattvachya topological gunadharmanche vaishishtya darshavinyasathi keela jaato, jeey (sarvasadharanpane), anukram vaishishtyikrut karoo shakat naheet (anukramanchya yaa kamataratemule anukramik jaaga aanee freckett-euresohan jagancha abhyas karnyas prerit hote). jaali filtershee jawalun sambandhit aahet, jyacha vapar anekada topologymadhehi keela jaato. pratyek jaale filtershee sambandhit asoo shakate aanee pratyek filter jaalishi sambandhit asoo shakate, jithe yaa sambandhit vastunche gunadharm jawalun jodalele asatat (adhik tapashilansathi topologymadhil filtervishayicha lekh pahaa). jaali thet anukramanche samanyikaran karate aanee teey sahasa anukramanpramanech vaparale jaau shakatat. parinami, jaali vaparnyasathi shikanyachi vakrata samanyatah filterchya tulanet khoopach qamee tivra asate, mhanunach anek ganitshastradnya, visheshtah vishleshak, filterpeksha tyanna pradhanya detat. tathapi, jaalipeksha filterche aanee visheshtah ultrafiltersche kahee mahattvache tantrik fayade aahet, jyamule shevati vishleshan aanee topologychya kshetrabaheril filterpeksha jaali khoop qamee vela samore yetat.
subnet mhanje nivval sootra 1 chey sootra 48 chya nirdeshit upasanchavarche nirbandh nahee, vyakhyesathi jodalele prushth pahaa.
jaalyanchi udaharane.
pratyek rikama nasalela purnapane aurder kelela sanch nirdeshit keela jaato. tyamule asha sanchavaril pratyek karya heey eka jaale asate. visheshtah, nehamichya kramane naisargik sankhya asha sanch tayar karatat aanee anukram heey naisargik sankhyanvaril karya asate, mhanoon pratyek kram nivval asato.
aanakhi eka mahattvache udaharan pudhilpramane aahe. topological spacemadhye bindu sootra _ 5 dile tarr sootra _ 50 heey sootra _ 51 asalelya serve shejaryancha sanch darshavu dya, nantar sootra _ 50 haah eka nirdeshit sanch aahe, jithe disha ulat samaveshanadvare dilee jaate, jenekarun sootra _ 53 jarr aanee fakt jarr sootra _ 54 sootra _ 56 sathee sootra _ 55 madhye asel tarr sootra _ 57 haah sootra _ 58 madheel bindu asel tarr sootra _ 59 heey nivval aahe. sootra _ 54 heey sootra _ 61 chya sandarbhat vadhat asatana, jaalitil guna sootra _ 57 heey sootra _ 8 chya qamee hott asalelya shejari padnyapurte maryadit asate, mhanoon antardnyanane sangayache tarr, aaplyala ashee kalpana yete qee sootra _ 57 neey kahee arthane sootra _ 5 kadey cull keela pahije. hii maryadit sankalpana aapan achook banavoo shakato.
anukramache subnet heey anivaryapane eka anukram asate. 
udaharanarth, sootra _ 66 aanee pratyek sootra _ 68 sathee sootra _ 67 asey asoo dya jenekarun sootra _ 69 haah sthir shoonya kram asel. 
sootra _ 70 heey nehamichya kram sootra _ 15 dware nirdeshit kele jaau dya aanee pratyek sootra _ 73 sathee sootra _ 72 asoo dya. 
sootra _ 75 heey sootra _ 76 chey kamal maryada asoo deuun sootra _ 74 paribhashit qaraa. 
nakasha sootra _ 74 heey eka kram rupankan aahe jyachi pratima tyachya kodomenmadhye sahadhi aahe aanee sootra _ 78 pratyek sutrasathi _ 73 dharan karate heey darshavite qee sootra _ 80 heey anukram sootra _ 41 chey upjal aahe (jethe heey upjal sootra _ 41 chey anugami nahee kaaran tyache kshetra eka aganit sanch asalyane too eka kram dekhil nahee).
jaalyanchya maryada.
jarr kahee sootra 29 astitvat asel tarr nivval sootra _ 83 heey kinwa sanch sootra _ 54 asey mhatale jaate, jenekarun sootra _ 87 asalelya pratyek sootra _ 30 sathee bindu sootra _ 88 asel. 
aanee jarr pratyek sutrasathi _ 29 asey kahee sootra _ 30 astitvat asel tarr teey sootra _ 92 aanee sootra _ 88 asey mhatale jaate. 
jarr teey jaale akheris (anukrame, saharthatah) tyaa binduchya pratyek shejari asel tarr eka bindula jaaleche (anukrame) aaa mhanatat.
jarr sootra _ 8 chya pratyek shejarchya sutrasathi _ 95 heey jaale varanvar sootra _ 97 madhye asel tarr spashtapane, bindu sootra _ 94 heey jaale kinwa jaale mhatale jaate. 
bindu sootra _ 94 laa sootra _ 2 madhye a kinwa nivval sootra _ 41 asey mhatale jaate jarr (aanee fakt jarr)
asha paristhitit, heey jaale nantar 'aanee' asehi mhatale jaate.
antardnyanane, nivval sootra 105 chey abhisaran mhanje moolye sootra 44 yete aanee aaplyala purese mothe sootra 108 sathee sootra 5 chya jevadhe jawal rahayache aahe tevadhe jawal rahate. 
bindu sootra _ 5 chya shejarchya pranalivar varr dilele udaharan jaale yaa vyakhyenusar kharokhar sootra _ 5 madhye ekatra yete.
maryadansathi sanketan
jarr nivval sootra _ 41 heey sootra _ 2 madhye bindu sootra _ 94 madhye ektrit jhaale tarr hii vastusthiti khalilpaiki konatihi lihoon vyakt kelee jaau shakateh
sootra _ 114
jethe jarr topological space sootra _ 2 sandarbhanusar spasht asel tarr 'sootra _ 2' madheel shabd vagalale jaau shakatat.
jarr sootra _ 2 madheel sootra _ 117 aanee sootra _ 2 madheel hii maryada adviteey asel (sootra _ 2 madheel advitiyatecha arth assa aahe qee jarr sootra _ 121 asey sootra _ 122 asel tarr aavashyakpane sootra _ 123) tarr hii vastusthiti lihoon darshavili jaau shakate.
sootra _ 124
jethe hasdorf jaaget baana sootra _ 125 chya jaagi samaan chinh vaparale jaate, pratyek jaalichi jastit jaast eka maryada asate tyamule hasdorf jaaget abhisari jaalichi maryada nehmich adviteey asate. 
kahee lekhak tyaeivaji sootra 117 chaa arth lavnyasathi 'sootra _ 126' heey sanket vapartat, jyamadhye maryada adviteey asane aavashyak aahe; tathapi, jarr heey sanket asha prakare paribhashit kele gele tarr samaan chinh sootra _ 128 aata sankramanshil sambandh darshavinyachya hamisathi nahee aanee mhanoon yapudhe samanata darshavat nahee. visheshtah, advitiyatechya garjeshivay, jarr sootra 129 vegale asel aanee jarr pratyek sootra 2 madheel sootra 41 chee maryada asel tarr sootra 123 chukiche asoonahi sootra 126 aanee sootra 133 (samaan chinh sootra 128 vaparoon) lihile jaau shakate.
aadhar aanee up-aadhar
sootra _ 2 vareel topologiesathi up-aadhar sootra _ 136 dile gele (jethe lakshat ghya qee topologiesathi pratyek aadhar dekhil eka up-aadhar aahe) aanee bindu sootra dile gele _ 138 sutratil nivval sootra _ 41 sootra _ 5 madhye ekatra yete jarr aanee jarr teey akheris sootra _ 51 chya pratyek shejarchya sutramadhye asel tarr heey vaishishtyikaran dilelya bindu sootra _ 51 chya shejarchya up-aadharanparyant (aanee tyachpramane shejarchya aadharanvar) vistarte.
metrik jaganmadhil abhisaran
samaja sootra 145 heey metrik space (kinwa seudometric space) aahe aanee sootra 2 heey metrik topologyne sampann aahe. 
jarr sootra _ 94 haah eka bindu asel aanee sootra _ 148 heey nivval asel, tarr sootra _ 145 madheel sootra _ 149 jarr aanee sootra _ 152 madheel sootra _ 151 asel tarr sootra _ 153 heey vastavik sankhyanche nivval asel. 
sadhya ingerjit, heey vaishishtyikaran asey mhanate qee jarr nivval aanee bindu yanchyatil antar shunyavar ekatra aale tarach nivval metrik spacemadhil bindumadhye ekatra yete. 
jarr sootra _ 154 hii eka manak jaaga (kinwa ardh-aakarachi jaaga) asel tarr sootra _ 154 madheel sootra _ 149 jarr aanee sootra _ 158 madheel sootra _ 157 asel tarr sootra _ 159 asel.
topological upasthananmadhil abhisaran
jarr sanch sootra _ 160 varr sootra _ 6 dware prerit upasthan topologyne sampann asel tarr sootra _ 2 madheel sootra _ 117 jarr aanee keval sootra _ 58 madheel sootra _ 117 asha prakare asel tarr, nivval sootra _ 41 dilelya bindu sootra _ 5 madhye ekatra yete qee nahee haah prashna yaa topological upasthan sootra _ 54 varr avalambun asato jyamadhye sootra _ 5 aanee nivval sootra _ 170 chee pratima (mhanje, bindu) asate.
cartasian utpadanat maryada.
jarr aanee pratyek prakshepanala maryada asel tarach utpadan jaaget nivval eka maryada asate.
spashtapane, sootra _ 171 heey topological spaces asoo dya, tyanche cartasian utpadan dya 
sootra _ 172
utpadanachya topologies, aanee pratyek nirdeshank sutrasathi _ 173 heey sootra _ 174 dware pramanit prakshepan darshavate
sootra _ 175
sootra _ 176 heey sootra _ 13 dware nirdeshit sootra _ 177 madheel nivval asoo dya aanee pratyek nirdeshank sutrasathi _ 179 asey asoo dya. 
sootra _ 180
sootra _ 181 laa sootra _ 182 madhye jodane" yacha parinam darshava, jyacha parinam nivval sootra _ 183 madhye hoto. 
karya rachanechya drushtine yaa vyakhyecha vichar karane kadhikadhi upayukt tharateh nivval sootra _ 184 heey prakshepan sootra _ 186 mhanjech sootra _ 187 saha nivval sootra _ 185 chya rachaneche samaan aahe.
jarr aanee pratyek nirdeshank sutrasathi _ 179 sootra _ 193 sootra _ 194 madhye sootra _ 195 madhye ektrit jhaale tarach, dilelya konatyahi bindu sutrasathi _ 188 nivval sootra _ 181 heey utpadan jaaga sootra _ 177 madheel sootra _ 190 madhye ektrit hote. 
aanee jevha jevha sootra _ 177 madheel sootra _ 190 varr nivval sootra _ 181 samooh hotat tevha pratyek nirdeshank sootra _ 201 sathee sootra _ 194 varr sootra _ 184 samooh, tathapi, ulat sarvasadharanpane tikat nahee. udaharanarth, samaja sootra 202 aanee sootra 203 heey anukram sootra 204 darshavate jeey sootra 205 aanee sootra 206 darmyan badalate, tarr sootra 207 aanee sootra 208 heey sootra 209 aanee sootra 210 yaa donhiche samooh bindu aahet, parantu sootra 212 haah sootra 181 chaa samooh bindu nahee kaaran sootra 215 varr kendrit trijya sootra 214 chya khulya chendumadhye eka bindu sootra dekhil nasate. 
tyconoffche pramey aanee nivadichya svayamsiddhateshi asalela sambandh
jarr konatehi sootra _ 217 dile gele nahee parantu pratyek sootra _ 179 sathee kahee sootra _ 219 astitvat aahe, jase qee sootra _ 221 madhye sootra _ 220 asel tarr sootra _ 222 dware paribhashit kelele tupal sootra _ 25 madhye sootra _ 181 chee maryada asel. 
tathapi, heey tupal sootra _ 190 astitvat aahe assa nishkarsh kadhanyasathi nivadicha svayamsiddha siddhant gruheet dharanyachi aavashyakta asoo shakate; kahee paristhitinmadhye nivadicha svayamsiddha siddhant aavashyak nasato, udaharanarth jevha sootra _ 226 maryadit asato kinwa jevha pratyek sootra _ 219 hii nivval sootra _ 184 chee maryada asate (kaaran nantar nivadanyasathi kahihi nasate), jeey udaharanarth ghadate, jevha pratyek sootra _ 221 hii eka hasdorff jaaga asate. jarr sootra _ 226 anant asel aanee sootra _ 239 rikame nasel, tarr andaj sootra _ 232 heey surjective nakashe aahet assa nishkarsh kadhanyasathi nivadicha svayamsiddha siddhant (sarvasadharanpane) aavashyak asel.
nivadicha svayamsiddha siddhant haah tyconoffchya prameyala samatulya aahe, jyamadhye asey mhatale aahe qee compact topological spaceschya konatyahi sankalanache utpadan compact aahe. 
parantu jarr pratyek sankshipt jaaga dekhil housedorf asel, tarr tyaeivaji tathakathit "compact hosdorf spesesasathi tyconoffche pramey" vaparale jaau shakate, jeey ultrafilter lemmachya samatulya aahe aanee nivadichya svayamsiddhatepeksha katekorpane kamkuvat aahe. 
jarr pratyek jaalyat abhisari subnet asel tarach jaaga sankshipt asel yaa vastusthitisaha varr dilelya nivval abhisaranache vaishishtya vaparoon tyconoffchya prameyachya donhi aavruttyanche sankshipt purave denyasathi jaalicha vapar keela jaau shakato.
jaalyache samooh bindu.
jarr aanee jarr tyacha upasanch sootra _ 51 madhye villin hoto tarach bindu sootra _ 94 haah dilelya jaalicha samooh bindu asato.
jarr sootra _ 83 heey sootra _ 2 madheel nivval asel tarr sootra _ 2 madheel sootra _ 41 chya serve samooh binduncha sanch samaan aahe
sootra _ 239
jethe pratyek sutrasathi sootra _ 240 _ 241 
jarr sootra _ 94 haah sootra _ 41 chya kahee subnetcha samooh bindu asel tarr sootra _ 5 dekhil sootra _ 170 chaa samooh bindu aahe.
ultranets.
jarr pratyek upasanch sutrasathi _ 248 sootra _ 41 heey shevati sootra _ 54 madhye asel kinwa sootra _ 41 heey shevati purak sootra _ 252 madhye asel tarr _ 2 yaa sanch sutratil nivval sootra _ 41 laa a kinwa an mhanatat. 
ultranetscha ultrafilters jawalcha sambandh aahe. 
pratyek sthir jaale heey eka ultranet asate. ultranetche pratyek subnet heey ultranet asate. pratyek jaalyat kahee subnet asate jeey ultranet asate. 
jarr sootra _ 83 heey sootra _ 2 madheel ultranet asel aanee sootra _ 255 heey karya asel tarr sootra _ 256 heey sootra _ 257 madheel ultranet aahe. 
sootra 138 dilele sootra 5 varr ultranet samooh jarr aanee fakt teey sootra 51 madhye ekatra yete
jaalichya maryadanchi udaharane.
kramachi pratyek maryada aanee karyachi maryada hii nivval maryada mhanoon samajali jaau shakate (khalee varnan kelyapramane). 
reeman samakalanchya mulyachya vyakhyecha arth reeman bereejchya nivval rakmechi maryada assa lavala jaau shakato, jethe nivval nirdeshit sanch haah samakalanchya madhyantaratil serve vibhajanancha sanch asato, joou anshatah samaveshadvare krambaddha keela jaato.
prototype sootra _ 262 saha serve karyanchya sanch sootra _ 261 chaa cartasian utpadan sootra _ 263 mhanoon arth kadha (tupal sootra _ 265 saha karya sootra _ 1 olakhoon aanee tyaulat) aanee tyala utpadan topologies pradan qaraa. sootra _ 261 vareel heey (utpadan) topology binduvyaj abhisaran topologisarkhech aahe. sootra _ 267 heey serve karyancha sanch soochit karuya, sootra _ 268 jeey sootra _ 214 chya barobariche aahet, jastit jaast maryadit anek bindu vagalata (mhanje, asey qee sanch sootra _ 270 maryadit aahe). mug sthir sootra _ 271 karya sootra _ 272 heey sootra _ 274 madheel sootra _ 267 chya bund honyashi sambandhit aahe, mhanje sootra _ 275 heey sootra _ 267 madhye jaali tayar karoon siddha kele jaeel jeey sootra _ 277 madhye ekatra yete, tathapi, sootra _ 267 madhye asey konatehi astitva nahee jeey sootra _ 279 madhye ekatra yete jyamule heey eka udaharan banate jithe (naun-sequence) jaali vaparali jaane aavashyak aahe kaaran ekte anukram ichchit nishkarshaparyant pohochu shakat naheet. jarr aanee serve sutransathi sootra 282 asel tarach hii binduvar tulana eka aanshik kram aahe, jeey sootra 284 laa nirdeshit sanch banavate, kaaran konatehi sootra dile asalyas, tyanche binduvar kiman sootra 286 heey sootra _ 267 chey aahe aanee sootra _ 288 aanee sootra _ 289 chey samadhan karate, hii aanshik kramvari olakh nakashyache sootra _ 290 (sootra _ 291 dware paribhashit) laa sootra _ 267-mulyayukt nivvalmadhye rupantarit karate. heey jaale _ 293 yaa sutramadhye bindupramane ekatra yete, jeey soochit karate qee sootra _ 293 heey sootra _ 297 madheel sootra _ 267 chya bund honyashi sambandhit aahe.
udaharane.
topological spacemadhil kram.
topological space formulamadhil anukram sootra _ 298 heey sootra _ 301 varr paribhashit kelelya sootra _ 2 madheel nivval manale jaau shakate.
jarr asey sootra _ 304 astitvat asel qee pratyek purnank sootra _ 305 sathee bindu sootra _ 306 sootra _ 58 madhye asel tarr nivval shevati sootra _ 2 chya upasanch sootra _ 54 madhye aahe.
mhanoon sootra 308 jarr aanee fakt jarr sootra 310 chya pratyek shejarchya sutrasathi _ 309 nivval shevati sootra 311 madhye asel
jarr aanee pratyek sutrasathi kahee purnank sootra _ 304 astitvat asel tarach nivval sootra _ 320 chya upasanch sootra _ 54 madhye asate, mhanje sootra _ 316 mhanje, jarr aanee fakt anukramache asankhya ghatak sootra _ 58 madhye asateel tarr bindu sootra _ 121 haah nivval samooh bindu asel jarr aanee sootra _ 320 chya pratyek shejarchya sootra _ 309 madhye anukramache anant ghatak asateel tarach.
metrik jagepasoon topological jageparyant karya karate.
metrik space formula _ 322 madhye bindu sootra _ 321 nishchit qaraa jyamadhye kiman done bindu asateel (udaharanarth sootra _ 323 aanee sootra _ 324 heey mool sootra asalele euclidean metrik, udaharanarth) aanee sootra _ 326 pasoonchya antaranusar sanch sootra _ 325 ulat nirdeshit qaraa, teey sootra _ 327 ghoshit karoon jarr aanee jagepasoon shabdant sootra _ 328 asel, tarr sambandh "kiman sootra _ 326 itake samaan antar asel", jenekarun yaa sambandhachya sandarbhat "purese mothe" mhanje "sootra _ 326 chya purese jawal". 
domain sootra _ 331 asalele konatehi karya dile tarr sootra _ 325 varr tyache nirbandh sootra _ 333 dware nirdeshit nivval mhanoon vihitpane arth lavala jaau shakato. 
nivval sootra _ 334 heey shevati topological space sootra _ 2 chya upasanch sootra _ 54 madhye asate, jarr aanee keval jarr kahee sootra _ 337 astitvat asel, jase qee pratyek sootra _ 338 samadhankarak sootra _ 339 sathee bindu sootra _ 340 heey sootra _ 58 madhye asel. 
jarr aanee nehamichya arthane sootra 345 asel tarach asey nivval sootra _ 1 heey sootra _ 2 madhye dilelya bindu sootra _ 217 madhye ekatra yete (mhanje sootra _ 310 chey pratyek shejari sootra _ 309 heey sootra _ 309 madhye aahe). 
nivval sootra _ 334 heey sootra _ 2 chya upasanch sootra _ 54 madhye asate, jarr aanee pratyek sootra _ 337 sathee sootra _ 355 saha kahee sootra _ 338 astitvat asel, jase qee sootra _ 340 sootra _ 58 madhye asel.
parinami, jarr aanee fakt jarr sootra _ 310 chya pratyek shejarchya sutrasathi _ 309 sathee nivval varanvar sootra _ 311 madhye asel, tarr bindu sootra _ 217 haah nivval sootra _ 1 chaa samooh bindu aahe.
changlya kramvarit asalelya sanchapasun topological jaaget karya karate.
maryada bindu sootra _ 364 saha suu-krambaddha sanch sootra _ 363 aanee sootra _ 366 pasoon topological space sootra _ 25 paryantche karya sootra _ 1 vicharat ghya heey karya sootra _ 368 varr nivval aahe.
jarr asey sootra _ 371 astitvat asel tarr teey shevati sootra _ 2 chya _ 309 yaa upasanch sutramadhye asel, jenekarun pratyek sootra _ 372 sathee bindu sootra _ 373 heey sootra _ 311 madhye asel.
mhanoon sootra _ 375 jarr aanee fakt jarr pratyek shejarchya sutrasathi sootra _ 310 sootra _ 1 chey sootra _ 309 shevati sootra _ 311 madhye asel
jarr aanee pratyek sutrasathi kahee sootra _ 372 astitvat asel tarach nivval sootra _ 1 heey sootra _ 2 chya upasanch sootra _ 309 madhye asate.
jarr aanee jarr sootra 389 chya pratyek shejarchya sutrasathi nivval varanvar sootra 311 madhye asel tarach bindu sootra _ 121 haah nivval sootra _ 1 chaa samooh bindu aahe.
pahile udaharan heey _ 391 sutrasah yache eka vishesh prakaran aahe.
kram-anukramit kram dekhil pahaa.
subnet.
netsasathi 'sequence' chey analogue mhanje 'subnet' chee kalpana aahe. subnetchya anek bhinn samatulya nasalelya vyakhya aahet aanee haah lekh stephen willardne 1970 madhye sadar keleli vyakhya vaparnar aahe, jee khalilpramane aaheh 
jarr sootra _ 83 aanee sootra _ 393 heey jaale asateel tarr sootra _ 394 laa a kinwa sootra _ 41 mhanatat, jarr tethe kram-jatan karnare nakasha sootra _ 396 astitvat asel, jase qee sootra _ 397 haah sootra _ 13 chaa saha-antim upasanch aahe aanee 
sootra _ 399 
nakasha sutrala _ 396 mhanatat aanee jarr jevha jevha sootra _ 327 asel tevha sootra _ 402 
397 heey sootra 13 madhye asanyacha arth assa aahe qee pratyek sootra 405 sathee kahee sootra 406 astitvat aahe, jase qee sootra 407
malmatta.
topologychya aksharshah serve sankalpana jaali aanee maryadannusar punha mandalya jaau shakatat. antardnyanala margadarshan karnyasathi heey upayukt tharoo shakate kaaran jaalichi maryada hii sankalpana anukramachya maryadeshi khoop samaan aahe. pudhil pramey aanee shabdasangrah heey samya drudh karnyas madat karatatah
topological gunadharmanchi vaishishtye.
bund sanch aanee bund karane
jarr aanee keval jarr sootra _ 54 madheel pratyek abhisari jaalicha pratyek maryada bindu sootra _ 58 chaa asel tarach upasanch sootra _ 408 sootra _ 2 madhye bund kele jaate. 
spashtapane, upasanch sootra _ 408 bund kele jaate, jarr aanee jevha jevha sootra _ 94 aanee sootra _ 414 heey sootra _ 54 madhye nivval moolya asel (mhanje serve sootra _ 29 sathee sootra _ 416), asey sootra _ 418 heey sootra _ 6 madhye asel, tevha aavashyakpane sootra _ 420
adhik samanyatah, jarr sootra _ 408 haah konatahi upasanch asel tarr bindu sootra _ 94 heey sootra _ 54 chya bund honyachya sthiteet asel, jarr aanee keval jarr sootra _ 54 madhye maryada sootra _ 94 saha nivval sootra _ 424 astitvat asel aanee pratyek nirdeshank sootra _ 241 sathee sootra _ 416 asel. 
topologyche khule sanch aanee vaishishtye
jarr aanee jarr sutratil konatehi jaale _ 430 heey sootra _ 58 chya binduvar ekatra yet nasel, tarr upasanch sootra _ 408 ughade asate, jarr aanee jarr sootra _ 54 chya ghatkamadhye rupantarit honara pratyek jaale akheris sootra _ 58 madhye samavisht asel tarach. 
heey 'open subset' chey heey vaishishtyapurn varnan aahe jeey jaalila topology vaishishtyikrut karnyas anumati dete. 
topologys bund upasanchandvare dekhil darshavili jaau shakatat kaaran jarr aanee keval jarr tyacha purak bund asel tarach sanch khula asato. tyamule jaalichya sandarbhat 'bund sanch' chee vaishishtye dekhil topologyche vaishishtyikaran karnyasathi vaparali jaau shakatat. 
satatya
topological spacemadhil function sootra _ 255 heey dilelya bindu sutravar nirantar asate, jarr aanee keval jarr tyachya kshetramadhil pratyek nivval sutrasathi _ 83, sootra _ 2 madheel sootra _ 117 asel tarr sootra _ 257 madheel sootra _ 440 asel. 
adhik sankshiptapane sangayache tarr, function sootra _ 255 heey satat asate jarr aanee jevha jevha sootra _ 2 madheel sootra _ 149 asel tevha sootra _ 257 madheel sootra _ 445 asel. 
sarvasadharanpane, jarr 'nett' yaa shabdachi jaaga 'anukram' neey ghetali tarr heey vidhan kharey tharnar nahee; mhanje, jarr sootra _ 2 hii pratham-mojanyayogya jaaga nasel (kinwa anukramik jaaga nasel) tarr keval naisargik sankhyanvyatirikt itar nirdeshit sanchanna parvanagi dene aavashyak aahe.
(sootra _ 448) 
sootra _ 1 heey bindu sootra _ 8 varr akhand asoo dya aanee sootra _ 83 heey asey nivval asoo dya qee sootra _ 452
mug sootra 454 chya pratyek khulya shejarchya sutrasathi _ 455 sootra 456 antargat tyache purvagrah sootra 5 chey shejari aahe (sootra 5 varr sootra 1 chya satatyane).
asha prakare sootra _ 460 chaa aateel bhag joou sootra _ 461 dware darshavila jaato too sootra _ 8 chaa eka khula parisar aahe aanee parinami sootra _ 41 akheris sootra _ 464 madhye aahe mhanoon sootra _ 465 akheris sootra _ 466 madhye aahe aanee asha prakare akheris sootra _ 467 madhye dekhil aahe jeey sootra _ 97 chaa upasanch aahe asha prakare sootra _ 469 aanee hii disha siddha jhaali aahe.
(sootra _ 470) 
sootra _ 5 haah assa bindu asoo dya qee pratyek nivval sootra _ 83 sathee asey sootra _ 473 sootra _ 474 aata samaja sootra _ 1 sootra _ 51 varr akhand nahee
mug formula 10 chey neberhood formula 95 aahe jyache formula 455 formula 460 antargat preimage heey formula 51 chey neberhood nahee kaaran formula 482 heey aavashyakpane formula 483 aahe aata containment preorders formula 5 chya khulya neberhoodcha sanch eka nirdeshit sanch aahe (kaaran asha pratyek done neberhoodche chhedanbindu dekhil formula 5 chey khule neberhood aahe).
aapan eka nivval sootra _ 83 tayar karato jenekarun sootra _ 5 chya pratyek khulya shejarisathi, jyacha nirdeshank sootra _ 488 sootra _ 44 aahe, haah yaa shejaril eka bindu aahe joou sootra _ 309 madhye nahee; sootra _ 309 madhye sootra _ 5 chaa konatahi khula shejari samavisht nahee yaa vastusthitivarun nehmich assa eka muddah yeto (kaaran gruheet dharoon, sootra _ 309 haah sootra _ 5 chaa shejari nahee).
heey asey sootra anusarate qee _ 495 heey sootra _ 97 madhye nahee
aata, sootra _ 8 chya pratyek khulya shejarchya sutrasathi _ 497 haah parisar nirdeshit sanchatil eka sadasya aahe jyacha nirdeshank aapan pratyek sootra _ 500 sathee sootra _ 499 darshavito, nirdeshit sanchatil sadasya jyacha nirdeshank sootra _ 501 aahe too sootra _ 497 madhye samavisht aahe; mhanoon sootra _ 503 asey sootra _ 452 aanee aaplya gruheet sootra _ 474 dware
parantu sootra _ 506 heey sootra _ 10 chey khule shejari aahe aanee tyamule sootra _ 495 heey akheris sootra _ 506 madhye aahe aanee mhanoon sootra _ 104 madhye dekhil aahe, jeey sootra _ 495 chya viruddha aahe aanee pratyek sootra _ 108 sathee sootra _ 95 madhye nahee.
haah eka virodhabhas aahe mhanoon sootra _ 1 heey sootra _ 51 varr satat asane aavashyak aahe, heey purava purn karate. 
sankshiptata
jarr aanee sootra _ 2 madheel pratyek nivval sootra _ 83 madhye sootra _ 25 madheel maryadesah subnet asel tarach heey bolzano-warestrass pramey aanee hein-borel pramey yanche samanyikaran mhanoon pahile jaau shakate.
(sootra _ 448) 
pratham, samaja sootra _ 2 sankshipt aahe. aaplyala khaleel nirikshanachi aavashyakta asel (maryadit chhedanbindu gunadharm pahaa). sootra _ 226 haah konatahi rikama sanch asoo dya aanee sootra _ 523 haah sootra _ 2 chya bund upasanchancha sangrah asoo dya, jase qee pratyek maryadit sutrasathi sootra _ 525, nantar sootra _ 527 dekhil. anyatha, sootra _ 528 heey sootra _ 2 sathee eka khule aavaran asel, jyamadhye sootra _ 25 chya sankshiptatechya viruddha konatehi maryadit up aavaran nasel.
pratyek sutrasathi _ 83 heey sootra _ 2 madhye sootra _ 533 dware nirdeshit nivval asoo dya _ 29 paribhashit qaraa
sootra _ 535
sangrah sootra _ 536 madhye assa gunadharm aahe qee pratyek maryadit upasanchalanat rikt nasalele chhedanbindu asate. tyamule vareel tippaninusar, aamchyakade teey aahe
sootra _ 537
aanee pudhil vibhagat dilelya puravyanusar haah tantotant sootra 170 chya samooh binduncha sanch aahe, too sootra 170 chya abhisari subnetchya maryadanchya sanchaitkach aahe, tyamule sootra 41 madhye abhisari subnet aahe.
(sootra _ 470) 
yaulat, samaja sootra _ 2 madheel pratyek jaalyat eka abhisari subnet aahe. virodhabhasasathi, sootra _ 543 heey konatehi maryadit upkoshth nasalele sootra _ 2 chey khule aavaran asoo dya. sootra 545 chaa vichar qaraa, heey lakshat ghya qee sootra 546 haah samaveshakhalil eka nirdeshit sanch aahe aanee pratyek sootra 547 sathee eka sootra 548 astitvat aahe, jase qee serve sutransathi sootra 549, nivval sootra 551 vicharat ghya, yaa jaalila abhisari subnet asoo shakat nahee, kaaran pratyek sutrasathi sootra 554 heey sootra 5 chey shejari aahe asey sootra 553 astitvat aahe; tathapi, serve sootra 556 sathee aamchyakade teey sootra 557 aahe, heey eka virodhabhas aahe aanee purava purn karate.
samooh aanee maryada bindu.
jaalichya samooh binduncha sanch tyachya abhisari subnetchya maryadanchya sanchaitka asato.
sootra _ 83 heey topological space formula _ 2 madhye nivval asoo dya (jethe nehmipramane sootra _ 13 svayamchalitapane nirdeshit sanch asalyache gruheet dharale jaate) aanee sootra _ 561 dekhil asoo dya jarr sootra _ 320 hii sootra _ 41 chya subnetchi maryada asel tarr sootra _ 320 haah sootra _ 170 chaa samooh bindu aahe.
yaulat, asey gruheet dhara qee sootra _ 320 haah sootra _ 170 chaa samooh bindu aahe.
sootra _ 568 heey jodyanche sootra _ 569 asoo dya, jethe sootra _ 95 heey sootra _ 2 madheel sootra _ 320 chey khule shejari aahe aanee sootra _ 29 asey aahe qee sootra _ 574 aahe.
tyanantar nakashanche sootra _ 575 mapping sootra _ 569 teey sootra _ 43 heey saha-antim asate.
shivay, formula _ 568 deuun utpad kram (formula _ 320 chey shejari samaveshadvare krambaddha kele jaatat) tyala eka nirdeshit sanch banavate aanee formula _ 581 dware paribhashit nivval formula _ 580 formula _ 582 madhye ekatra yete.
jarr aanee keval tyachya serve subnetna maryada asateel tarach netchi maryada asate. asha paristhitit, jaalichi pratyek maryada hii pratyek subnetchi maryada dekhil asate.
itar malmatta.
sarvasadharanpane, avkash sootra _ 2 madheel jaalichi maryada ekapeksha jaast asoo shakate, parantu sootra _ 2 hii eka housedorf jaaga asalyas, jaalichi maryada, tee astitvat asalyas, adviteey aahe. yaulat, jarr sootra _ 2 heey hasdorf nasel, tarr sootra _ 2 varr done bhinn maryadansah eka jaale astitvat aahe. asha prakare maryadeche vaishishtya mhanje jagevaril hosdorf sthitee aanee kharokharach hii vyakhya mhanoon ghetali jaau shakate. haah parinam dishadarshak sthitivar avalambun asato; samanya purvaniyojanedware kinwa aanshik kramane anukramit kelelya sanchamadhye agadi hosdorf jagetahi vishisht maryada bindu asoo shakatat.
kachee jaali.
eksaman jaganvar paribhashit kelelya jaalimadhye eka kapati jaale, kapati anukramachi kalpana samanya karate.
nivval sootra _ 83 heey asey aahe qee jarr pratyek dall sootra _ 309 sathee sootra _ 17 astitvat asel, mhanje serve sutrasathi _ 590 sootra _ 591 heey sootra _ 311 chey sadasya aahe, adhik samanyatah, kachee spacemadhye, jarr netdware tayar kelele filter kachee filter asel tarr nivval sootra _ 41 kachee aahe.
jarr pratyek kachee jaale kahee binduparyant ekatra yete tarr topological vector space (tvs) mhanatat. jarr aanee pratyek kachee kram kahee binduvar (jyala gunadharm mhanatat) ekatra yeto tarach, eka samanya jaaga, jee eka vishesh prakarchi topological vector jaaga aahe, tee eka sampoorn durdarshan (samatulya, eka banach jaaga) asate. jaree samanya jaganchya paripoornateche varnan karnyasathi kachee jaalichi aavashyakta nasali taree, adhik samanya (bahudha gair-samanya) topological vector jaganchya sampurnateche varnan karnyasathi tyanchi aavashyakta asate.
filtershee sambandhit.
filter hii topologymadhil aanakhi eka kalpana aahe jee samanya topological jaganmadhye abhisaranasathi samanya vyakhyela anumati dete. donhi kalpana yaa arthane samatulya aahet qee tyaa ektrikaranachi samaan sankalpana detat. adhik visheshtah, pratyek filter besasathi eka tayar keela jaau shakato aanee filter besache abhisaran sambandhit jaaleche abhisaran soochit karate-aanee ulatya margane (pratyek jaalisathi eka filter bases aahe aanee jaaleche abhisaran filter besache abhisaran soochit karate). udaharanarth, sootra _ 2 madheel konatehi nivval sootra _ 105 heey shepatichya sootra _ 596 chaa filter bases prerit karate, jethe yaa filter besdware tayar kelelya sootra _ 2 madheel filterla nett mhanatat. heey patravyavahar konatyahi prameyala, jeey eka sankalpanesah siddha kele jaau shakate, jagepasoon sankalpanesah siddha karnyas anumati dete. udaharanarth, eka topological jagepasoon jagepasoon thikani karyachi satatyata ektar domenmadhil jaaleche abhisaran, jeey kodomenmadhil sambandhit jaaleche abhisaran soochit karate, kinwa filter bases tyach vidhanadvare darshavili jaau shakate.
robbert jee. bartlecha assa yuktivad aahe qee tyanchi samanata asoonahi, donhi sankalpana asane upayukt aahe. tyancha assa yuktivad aahe qee naisargik purave aanee vyakhya, visheshtah vishleshanat samanya asalelya anukramik ghatakancha vapar vapar, anukramanshi sadrushya sadhanyasathi jaali hii anukramansarkhi pureshi aahet, tarr filter bijganitiy topologymadhye sarvat upayukt aahet. konatyahi paristhitit, samanya topologymadhye vividh pramey siddha karnyasathi yaa doghancha sanyog kassa keela jaau shakato heey too dakhavato.
maryada jaast.
vastavik sankhyanchya nivval maryadepeksha uchch aanee nimn maryada hii anukramanpramanech paribhashit kelee jaau shakate. kahee lekhak vastavik reshepeksha adhik samanya sanrachanansah dekhil kaam karatat, jase qee sampoorn jaali.
nivval sutrasathi _ 35 putt
sootra _ 599
vastavik sankhyanchya nivval maryadepeksha varchi maryada hii anukramanchya babatit samaan anek gunadharm aahet. udaharanarth,
sootra _ 600
jethe jevha jevha eka jaali abhisari asate tevha samanata tikate.

