
vitni topologykal

ganithashaasthrathil, prathyekichu deferential topology, functional analysis, singulaaritti sidhaantham ennivayil, vaitni topologykal rund minusamarnna maanifoldukal thammilulla minusamarnna maappingukalude koottatthil nirvachichirikkunna topologykalude ananthamaaya kudumbamaanu. american ganithashaasthranjanaaya hasler vitniyude perilaanu iva ariyappedunnathu.
nirmmaanam.
m", "n" enniva yadhaarthavum minusamarnnathumaaya rund manifoldukalaayirikkatte. koodaathe, c φ ("m", "n") ennathu "m" num "n" num edayilulla minusamarnna maappingukalude idathe soochippikkatte. c φ enna notteshan ardhamaakkunnathu mappingukal ananthamaayi deferentiable aanu, i.iii. ellah oordarukaludeyum bhagika derivateevukal nilavilundu, avaa thudarchayaayirikkunnu.
vitni sii "keey"-topology.
chila poornnasamkhyakalkku, "m" num "n" num edayilulla maappingukalude "k"-jettu speysine soochippikkaan "k" ("m", "n") ennu karuthuka. jettu space oru minusamarnna ghadana nalkaam (i.iii. oru sii φ manifold enna ghadana) athine oru topological space akkunnu. c φ ("m", "n") ennathil oru topology nirvachikkaan eee topology upayogikkunnu.
oru nischitha poornnasamkhyaykku oru thuranna upavibhagam pariganikkukayum inipparayunnava "ska" ("yuu") kondu soochippikkukayum cheyyuka
ska" ("yuu") enna settukal c ≤ ("m", "n") ennathil vaitni c "k"-topology oru adisthaanamaanu.
vitni sii φ-topology.
oro choysinum, vaitni sii "keey"-topology sii φ ("emm", "enn")-nuu oru topology nalkunnu; mattoru vidhathil paranjaal vaitni sii "keey"-topology sii φ ("emm", "enn")-nte eathu upavibhagangalaanu ooppan settukalennu nammodu parayunnu. vaitni sii "keey"-topologyumaayi bandhappett sii φ ("emm", "enn") nte ooppan subsettukalude koottatthe namukk dablyu "keey" kondu soochippikkaam. appol vitni sii φ-topology topologyayi nirvachikkunnu, athinte adisthaanam "dablyu" nalkunnu, avide
alavukal.
c φ ("m", "n") nuu ananthamaaya alavukalundennum athesamayam j "k" ("m", "n") nuu parimithamaaya alavukalundennum shradhikkuka. vaasthavathil, jeey "keey" ("emm", "enn") oru yadhaartha, parimitha-dimensional maanifoldaanu. ithu kaanunnathin, yadhaartha gunakangalulla polynomialukalude space, paramavadhi "k" kramatthilulla "m" veriyabilukalilum poojyathe sthiramaaya padamaayum soochippikkaam. ithu alavulla oru yadhaartha vekttar space aanu
appol vekttar speysukalude staanderd sidhaantham anusarichu ezhuthuka, athupole thanne oru yadhaartha, parimitha-dimensional manifold aanu. aduthathaayi, nirvachikkuka
bii" "keey" "emm", "enn" "enna alavine soochippikkaan" bii "upayogikkunnathu, nammal athu kaanunnu, athupole thanne oru yadhaartha, parimitha-dimensional manifold.
vaasthavathil, "m", "n" ennivaykku yathaakramam "m", "n" alavukalundenkil
topology.
vaitni sii ≤-topology nalkiyaal, c ≤ ("m", "n") enna space oru bair space aanu, i.iii. avasheshikkunna oro koottavum saandramaanu.

