
sub bays

topologyil, topology formula 2 ulla oru topological space formula _ 1-nulla oru subbase (allengil subbase, prebase, prebase) formula _ 2-nte oru sub kalakshan formulayanu, ithu formula _ 5 srishtikkunnu, athaayath formula _ 2 aanu formula _ 3 ooppan settukalaayi adangiyirikkunna ettavum cheria topology. chila rachayithakkal alpam vyathyasthamaaya oru nirvachanam upayogikkunnu, koodaathe nirvachanathinu samanamaaya mattu upayogapradamaaya soothravaakyangalum undu; iva chuvade charchacheyyunnu.
nirvachanam.
formula _ 1 topology formula _ 9 ulla oru topological space ayirikkatte, formula _ 2-nte oru upavibhagam saadhaaranayaayi formula _ 2-nte oru upavibhaga formula _ 3 aayi nirvachikkappedunnu, ithu inipparayunna rund thulya vyavasthakalil onnu thripthippeduthunnu
power settu formula 34-nte upavibhaga soothravaakyamaaya 33-nuu oru upavibhagamaayi soothravaakyam 33 ulla oru savisheshamaaya topology undu. prathyekichum, formula _ 33 adangiya formula _ 1 leeii ellah topologykaludeyum intersection eee vyavasthaye thripthippeduthunnu. ennirunnaalum, pothuve, oru nischitha topologyil savisheshamaaya upavibhagangalonnumilla.
athinaal, namukk oru nischitha topologyil ninnu aarambhich aaa topolajiyilekkulla upavibhagangal kandethaam, koodaathe power settu formula _ 38-nte ekapaksheeyamaaya upavibhaagathil ninnu aarambhich aaa upavibhagam srishtikkunna topology roopappeduthaam. mukalil paranja thulyamaaya nirvachanam namukk swathanthramaayi upayogikkaan kazhiyum; theerchayaayum, pala kesukalilum, rund vyavasthakalil onnu mattethinekkaal upayogapradamaanu.
badal nirvachanam.
saadhaaranayaayi, sabbesinte alpam vyathyasthamaaya oru nirvachanam nalkiyittundu, eee saahacharyathil subbase formula _ 39 cover formula _ 40 aavasyamaanu, formula _ 1 ennathu formula _ 42 ll adangiyirikkunna ellah settukaludeyum samyojanamaanu, ithinartham nirvachanathil shoonyamaaya intersectionukalude upayogathekkurichu aashayakkuzhappamundakilla ennaanu.
ennirunnaalum, eee nirvachanam ellaayppozhum mukalil paranja rund nirvachanangalkk thulyamalla. mattoru vidhathil paranjaal, formula _ 46 adangiya ettavum cheria topologyaanu formula _ 2 enna reethiyil oru upasett formula _ 44 ulla topological space formula _ 43 nilavilundu, ennittum formula _ 39 formula _ 1 ulkkollunnilla (atharamoru udaaharanam chuvade nalkiyirikkunnu). praayogikamaayi, ithu oru apoorva sambhavamaan; udha.jii. kuranjathu rund pooyintukalullathum tii 1 seppareshan axium thripthippeduthunnathumaa oru spacinte oru subbase aaa spacinte oru cover aayirikkanam, ennaal chuvade kaanunnathupole, ningalkk alexander subbase sidhaantham theliyikkanamenkil, ningal aaa formula _ 39 cover formula _ 40 ennu anumaanikkanam.
udaaharanagal.
eathenkilum upavibhaga soothravaakyam _ 51 (shoonyamaaya settu soothravaakyam _ 52 ulppede) srishtikkunna topology nissara topology soothravaakyam _ 53 nuu thulyamaanu.
formula _ 2 formula _ 1-leeii topology formula _ 39 formula _ 2-nte adisthaanavumaanenkil formula _ 39 srishtikkunna topology formula _ 9 aanu, athinaal oru topology formula _ 2-nulla eathu adisthaana formula formula _ 9-nte oru upavibhagamaanu. 
formula _ 63 formula _ 2-nte eathenkilum upavibhaagamaanengil formula _ 63 srishtikkunna topology formula _ 9-nte oru upavibhagamaayirikkum.
yadhaartha samkhyakalude soothravaakyathile saadhaarana topologyil, formula _ 68 allengil formula _ 69 enna formulayude ellah ardha-ananthamaaya thuranna edavelakalum ulkkollunna oru upavibhaagamund, avide formula _ 70 umm formula _ 71 umm yadhaartha samkhyakalaanu. iva orumichu saadhaarana topology srishtikkunnu, kaaranam formula _ 73-nte intersection formula _ 72 saadhaarana topology srishtikkunnu. soothravaakyam _ 70 umm soothravaakyam _ 71 umm yukthisahamaya upakudumbathe eduthaanu randaamathe upavibhagam roopappedunnathu. randaamathe subbase saadhaarana topology srishtikkunnu, kaaranam formula _ 77 formula _ 71 yukthisahamaya ooppan interwell formula _ 76 saadhaarana euclidean topology adisthaanamaanu.
formula _ 70 oru yadhaartha samkhyayaayirikke, phoom formula _ 68-nte ellah ardha-ananthamaaya thuranna edavelakalum ulkkollunna upavibhagam saadhaarana topology srishtikkunnilla. thathfalamaayundaakunna topology tii1 verthirikkal soothravaakyathe thripthippeduthunnilla, kaaranam soothravaakyam _ 81 aanenkil soothravaakyam _ 71 adangiyirikkunna oro ooppan settilum soothravaakyam _ 83 adangiyirikkunnu.
oro soothravaakyathinum oru topology ulla formula 85 enna functionukalude oru kudumbam nirvachicha formula _ 1-leeii prarambha topology, oro soothravaakyavum _ 88 thudarchayulla formula _ 1-leeii ettavum parukkan topologyaanu. ooppan settukalude vipareeta chithrangalude adisthaanathil thudarcha nirvachikkaan kazhiyumennathinaal, formula _ 1 leeii prarambha topology ellah soothravaakyangalum _ 90 eduthaanu nalkunnathu ennaanu ithinartham.
ivide formula _ 91 oru upavibhagamaayi formula _ 92-nte ellah ooppan upavibhaagangalilum vyaapikkunnu.
prarambha topology rund pradhana prathyeka kesukal uthppanna topology aanu, avide functionukalude kudumbam uthppannathil ninnu oro gadakathilekkum projakshanukalude koottamaanu, koodaathe kudumbam oru function maathram ulkkollunna subspace topology, inclusion mapp.
formula _ 1 muthal formula _ 94 vareyulla thudarchayaaya functionukalude sthalathe compact-ooppan topologyil oru sabbesinu oru koottam functionukal undu.
soothravaakyam _ 95
ivide formula _ 96 kompaakt aanu, formula _ 91 formula _ 98 nte oru ooppan upavibhagamaanu.
aaa formula _ 43 rando athiladhikamo moolakangal adangiya formula _ 1 ulla oru hosdorff topological space aanennu karuthuka (udaaharanathinu, euclidean toppolajiyodukoodiya formula _ 101). formula _ 102 ennathu formula _ 43-nte eathenkilum shoonyamallaatha upavibhagamaayirikkatte (udaaharanathinu, formula _ 94 formula _ 67-leeii shoonyamallaatha parimithikalillaattha thuranna edavelayaayirikkaam) koodaathe formula _ 106 formula _ 94-leeii subuspace topology soochippikkatte, formula _ 94 formula _ 43-ll ninnu paarambaryamaayi labhikkunnu (athinaal formula _ 110). appol formula _ 1-ll formula _ 106 srishtikkunna topology union formula _ 113-nuu thulyamaanu (oru vishadeekaranathinaayi eee adikkurippu kaanuka), ivide formula _ 114 (formula _ 43 haddorf aayathinal, formula _ 116 aanenkil maathram samathvam nilanilkkum). formula _ 94 formula _ 22-nte sariyaaya upavibhaagamaanengil formula _ 113 aanu "formula _ 1-leeii ettavum cheria topology" formula _ 106 adangiyittundengilum formula _ 106 formula _ 1 ulkkollunnilla (athaayath, union formula _ 124 formula _ 1-nte sariyaaya upavibhagamaanu).
sub baseukal upayogichu phalangal.
oru functionte thudarcha shreniyude oru upavibhaagathil maathrame parishodhikkendathulloo ennathaanu sub besukalekkurichulla oru nalla vastutha. athaayath, formula _ 126 topological speysukalkkidayilulla oru bhoopadamaanenkil, formula _ 39 formula _ 128-nte oru upavibhaagamaanengil, formula _ 130 oro formulakkum formula _ 1-ll thurannirichaal maathrame formula _ 126 thudarchayaayulloo. 
oru nettu (allengil sequence) formula _ 133 oru pooyint formula _ 26 aayi othucherunnu, formula _ 26-nte oro adisthaana ayalpakkathilum mathiyaya valia formula _ 137-nuu ellah formulakalum _ 136 adangiyirikkukayaanenkil maathram
alexander subbase sidhaantham.
jaims wadell alexander randaaman kaaranamulla upavibhagangalekkurichula oru pradhana falamaanu alexander subbase sidhaantham. adisthaana (sub basic kavarukalekkaal) ooppan kavarukalkku samanamaaya phalam theliyikkaan valare eluppamaanu.
eee sidhaanthathinu vipareethamaayi 144 enna soothravaakyam upayogichukondu ithu theliyikkappedunnu (oro topology svayam oru upavibhagamaayathinaal). 
vairudhyathinaayi space formula _ 1 kompaakt allennu karuthuka (athinaal formula _ 1 oru ananthamaaya koottamaanu), ennittum formula _ 63 ll ninnulla oro subbasic kavarinum oru parimithamaaya upavibhaagamund. 
formula _ 153 formula _ 40-nte parimithamaaya upavibhagangalonnumillaath formula _ 1-nte ellah thuranna kavarukaludeyum koottatthe soochippikkatte. 
upavibhagam ulppeduthal vazhi bhaagikamaayi formula 153 oordar cheyyuka, koodaathe formula _ 158-nte paramavadhi gadakamaaya oru moolaka formula _ 157 kandethaan soninte lemma upayogikkuka. 
ithu shradhikkuka
soothravaakyam _ 176 ennathu soothravaakyam _ 40-nte oru cover aanennu kaanichu nammal aarambikkum. 
aaa soothravaakyam _ 176 ennathu soothravaakyathinte oru cover aanennu karuthuka, ithu formula _ 182 nte gadakangalaal formula _ 176 formula _ 1 nte cover aanennu prathyekamaayi soochippikkunnu. 
soothravaakyam _ 63-nekkurichulla sidhaanthathinte sidhaantham soochippikkunnathu soothravaakyam _ 226-nte oru parimithamaaya upavibhagam nilavilundennaanu, athu soothravaakyam _ 22 ulkkollunnu, ithu oresamayam soothravaakyam _ 161-nte gadakangalaal soothravaakyam _ 1-nte parimithamaaya upavibhagamaayirikkum (soothravaakyam _ 188 muthal). 
ennaal ithu formula 159 nuu virudhamaanu, ithu formula 176 formula 40 ulkkollunnillenn theliyikkunnu.
soothravaakyam _ 176 formula _ 22 ulkkollathathinaal, formula _ 176 ulkkollaatha chila soothravaakyam _ 194 nilavilundu (athaayath, formula _ 26 formula _ 176-nte oru gadakathilum adangiyilla). 
ennaal formula _ 161 formula _ 22 ulkkollunnathinaal formula _ 201 polulla chila formula _ 200 nilavilundu. 
formula _ 63 ennathu formula _ 1 nte topology srishtikkunna oru upavibhagamaayathinaal, formula _ 204 srishticha topology nirvachanathil ninnu 205 enna upa basic ooppan settu formulayude oru parimithamaaya shekharam undaayirikkanam.
soothravaakyam _ 206
oro soothravaakyathinum _ 208 enna soothravaakyam vairudhyathiloode nammal ippol kanikkum. 
formula _ 209 aaa formula _ 210 aayirunnengil formula _ 212 umm formula _ 211 aanenkil formula _ 26 engane thiranjedutthu ennathinu virudhamaaya formula _ 214 kondu ulkkollunnu ennu formula _ 212 soochippikkum (formula _ 26 thiranjedutthathu formula _ 176 kondu ulppeduthaatheyaanennu oorkkuka).
neratthe soochippichathupole, formula 153 leeii formula 161 nte paramavadhi soochippikkunnathu oro formulakkum _ 220 nuu formula 161 nte oru parimithamaaya upasett formula _ 221 nilavilundennaanu, atharam formula 223 formula 40 nte parimithamaaya cover roopappeduthunnu. 
nirvachikkuka
soothravaakyam _ 225
ithu _ 226 enna soothravaakyathinte parimithamaaya upavibhagamaanu 
oro soothravaakyathinum _ 220 soothravaakyam _ 228 ennathu soothravaakyathinte oru parimitha cover aanennu nireekshikkuka, athinaal namukk oro soothravaakyathinum _ 221 enna soothravaakyam _ 231 upayogichu maattisthaapikkaam.
formula _ 232 formula _ 233 (ithu formula _ 1 nte oru ooppan upavibhagamaanu) leeii ellah settukaludeyum samyojanathe soochippikkatte, formula _ 235 formula _ 40 leeii formula _ 232 nte poorakatthe soochippikkatte. 
formula 241 aanenkil maathram eathu upasett formulakkum _ 238 formula _ 239 formula _ 1 ulkkollunnuvennu nireekshikkuka. 
prathyekichum, oro formulakkum _ 220 enna formula _ 228 formula _ 1 ulkkollunnu enna vastutha soochippikkunnathu formula _ 245 ennaanu. 
formula _ 209 ekapaksheeyamaayathinaal, namukk formula _ 247 undu. 
aaa formula _ 248 oormmikkumbol namukk formula _ 249 undu, ithu formula _ 250 nuu thulyamaanu, ithu formula _ 40 nte cover aanu. 
koodaathe, formula _ 250 ennathu formula _ 254 ulla formula _ 1-nte parimithamaaya kavaraanu. 
athinaal formula _ 161 nuu formula _ 22 nte parimithamaaya upavibhaagamund, ithu formula _ 257 enna vasthuthaykk virudhamaanu. 
athinaal, formula _ 1 kompaakt allenna yadhaartha anumaanam thettayirikkanam, ithu formula _ 1 kompaakt aanennu theliyikkunnu. soothravaakyam _ 260
eee thelivu soninte lemma upayogikkunnundengil, thelivinu thiranjedukkaan poornnamaaya shakti aavasyamilla. 
pakaram, ithu intermediat ultrafilter thathwathe aasrayichirikkunnu.
mukalilulla soothravaakyam _ 67-nte upabesinoppam eee sidhaantham upayogichu, soothravaakyathile parimithamaaya clossed intervalukal kompaakt aanennathin oraalkk valare eluppathil thelivu nalkaan kazhiyum. 
kooduthal pothuve, shoonyamallaatha compact speysukalude uthppannam othukkamullathaanennu parayunna tyconophinte sidhaanthathinu alexander subbase sidhaantham upayogikkukayaanenkil oru hraswa thelivu undu.
263 enna soothravaakyathile uthppanna topologyil, nirvachanam anusarichu, oru gadakathil oru ooppan settinte vipareeta pravachanangalaya "silindar" settukal adangiya oru subbase undu. 
parimithamaaya upavibhagamillaatha uthppannathinte oru kudumba soothravaakyam 264 nalkiyaal, namukk soothravaakyam 265 oru nischitha ghadakam sthalavumaayi bandhapetta silindar settukal krithyamaayi ulkkollunna upakudumbangalayi vibhajikkaam. 
anumaanichaal, formula _ 266 aanenkil formula _ 267 nuu oru parimithamaaya upavibhaagamund. 
silindar settukalaayathinaal, ithinartham _ 268 enna formula avayude projakshanukalkku parimithamaaya upavibhagamilla, oro formula _ 268 kompaakt aayathinal, formula _ 272-lekkulla _ 267 enna formulayude projakshanukalaal ulppedaatha oru pooyint formula _ 270 namukk kandethaan kazhiyum, ennaal formula _ 273 formula _ 274 formula _ 260 kondu ulppedunnilla.
shradhikkuka, avasana ghattathil nammal formula _ 276-nte nilanilpp urappakkaan choys oof choys (ithu yadhaarthathil soonn lemmaic thulyamaanu) parokshamaayi upayogichu.