Baza okoliv u tochci i sistema okoliv bazovi ponyattya u zagalnij topologiyi za dopomogoyu yakih mozhna dati oznachennya topologichnogo prostoru ekvivalentni standartnim oznachennyam za dopomogoyu vidkritih mnozhin Za dopomogoyu sistem chi baz okoliv dayetsya oznachennya neperervnoyi u tochci funkciyi Zmist 1 Oznachennya 2 Prikladi 3 Vlastivosti 4 Kardinalni funkciyi 5 Div takozh 6 DzherelaOznachennya RedaguvatiNehaj X t displaystyle X tau nbsp topologichnij prostir i x X displaystyle x in X nbsp Mnozhina vsih okoliv ne obov yazkovo vidkritih tochki x displaystyle x nbsp nazivayetsya sistemoyu okoliv u tochci x displaystyle x nbsp Dlya neyi vikoristovuyetsya poznachennya V x displaystyle mathcal V x nbsp Mnozhina B x V x displaystyle mathcal B x subset mathcal V x nbsp okoliv tochki x displaystyle x nbsp nazivayetsya bazoyu okoliv u tochci x displaystyle x nbsp abo fundamentalnoyu sistemoyu okoliv tochki x displaystyle x nbsp yaksho U V x V B x V U displaystyle big forall U in mathcal V x big big exists V in mathcal B x in V subseteq U big nbsp U podibnij sposib takozh mozhna dati oznachennya sistem i baz okoliv dovilnoyi pidmnozhini topologichnogo prostoru Prikladi RedaguvatiSistema okoliv tochki x displaystyle x nbsp ye takozh bazoyu okoliv u cij tochci Yaksho X displaystyle X nbsp ye diskretnim prostorom to B d x x displaystyle mathcal B d x big x big nbsp odnoelementna mnozhina ye bazoyu okoliv u x X displaystyle x in X nbsp Yaksho X displaystyle X nbsp ye antidiskretnim prostorom to B a x X displaystyle mathcal B a x X nbsp ye bazoyu okoliv u x X displaystyle x in X nbsp Yaksho X displaystyle X nbsp ye metrichnim prostorom z metrikoyu d displaystyle d nbsp i dlya tochki x X displaystyle x in X nbsp i chisla r gt 0 displaystyle r gt 0 nbsp poznachimo B x r y X d x y lt r displaystyle B x r y in X d x y lt r nbsp to todi sim ya B x 1 n n 1 2 3 displaystyle B x 1 n n 1 2 3 ldots nbsp ye bazoyu okoliv u x displaystyle x nbsp Vlastivosti RedaguvatiTut yak i u statti Okil okolom tochki nazivayetsya mnozhina sho mistit vidkritu mnozhinu elementom yakoyi ye dana tochka tobto okoli ne obov yazkovo ye vidkritimi mnozhinami Nehaj V x x X displaystyle mathcal V x x in X nbsp ye sistemoyu okoliv topologichnogo prostoru X displaystyle X nbsp Todi vikonuyutsya taki vlastivosti Dlya kozhnogo x X displaystyle x in X nbsp V x displaystyle mathcal V x neq emptyset nbsp i dlya kozhnogo U V x displaystyle U in mathcal V x nbsp x U displaystyle x in U nbsp Yaksho U V x displaystyle U in mathcal V x nbsp i U V X displaystyle U subset V subset X nbsp to takozh V V x displaystyle V in mathcal V x nbsp Yaksho U V x displaystyle U in mathcal V x nbsp to isnuye U V V x displaystyle U supset V in mathcal V x nbsp takij sho U V y displaystyle U in mathcal V y nbsp dlya kozhnoyi tochki y V displaystyle y in V nbsp Peretin skinchennoyi kilkosti elementiv V x displaystyle mathcal V x nbsp tezh ye elementom V x displaystyle mathcal V x nbsp Pershi dvi vlastivosti viplivayut iz oznachennya okolu tochki chetverta iz togo sho peretin skinchennoyi kilkosti vidkritih mnozhin ye vidkritoyu mnozhinoyu i z togo faktu sho za oznachennyam kozhen okil mistit vidkritij okil U tretij vlastivosti za mnozhinu V displaystyle V nbsp mozhna vzyati dovilnij okil yakij isnuye za oznachennyam Vlastivist oderzhuyetsya z togo faktu sho vidkrita mnozhina ye okolom vsih svoyih tochok i tomu dovilna mnozhina sho yiyi mistit tezh ye okolom vsih yiyi tochok dd Navpaki pripustimo sho X displaystyle X nbsp ye nepustoyu mnozhinoyu i V x x X displaystyle mathcal V x x in X nbsp ye sistemoyu simej pidmnozhin X displaystyle X nbsp sho zadovolnyayut vlastivosti 1 4 Nehaj t displaystyle tau nbsp sim ya vsih pidmnozhin X displaystyle X nbsp takih sho U V x displaystyle U in mathcal V x nbsp dlya vsih x U displaystyle x in U nbsp Todi t displaystyle tau nbsp ye topologiyeyu na X displaystyle X nbsp i V x x X displaystyle mathcal V x x in X nbsp ye sistemoyu okoliv dlya ciyeyi topologiyi Topologiya t displaystyle tau nbsp nazivayetsya topologiyeyu porodzhenoyu sistemoyu okoliv B x x X displaystyle mathcal B x x in X nbsp Takim chinom sistema okoliv mozhe buti odnim iz sposobiv zadannya topologiyi na mnozhini Ochevidno sho pusta mnozhina i ves prostir nalezhat t displaystyle tau nbsp Dlya dovilnoyi sim yi mnozhin iz t displaystyle tau nbsp yih ob yednannya mistit kozhnu iz cih mnozhin i tomu zgidno drugoyi vlastivosti ye okolom vsih svoyih tochok Tobto ob yednannya dovilnoyi sim yi mnozhin iz t displaystyle tau nbsp tezh nalezhit t displaystyle tau nbsp Dlya skinchennoyi sim yi mnozhin iz t displaystyle tau nbsp kozhna z cih mnozhin ye okolom kozhnoyi z tochok yih peretinu i tomu dlya kozhnoyi z cih tochok peretin mnozhin ye okolom zgidno chetvertoyi vlastivosti Tomu peretin skinchennoyi sim yi pidmnozhin z t displaystyle tau nbsp tezh nalezhit t displaystyle tau nbsp i tomu t displaystyle tau nbsp ye topologiyeyu Zgidno drugoyi vlastivosti kozhen okil tochki x X displaystyle x in X nbsp nalezhit V x displaystyle mathcal V x nbsp Navpaki nehaj V V x displaystyle V in mathcal V x nbsp i U displaystyle U nbsp mnozhina tochok y displaystyle y nbsp dlya yakih V V y displaystyle V in mathcal V y nbsp Ochevidno sho x U displaystyle x in U nbsp i U V displaystyle U supset V nbsp Dovedemo sho mnozhina U displaystyle U nbsp ye vidkritoyu u topologiyi t displaystyle tau nbsp Nehaj y U displaystyle y in U nbsp Todi zgidno vlastivosti 3 isnuye taka mnozhina W V y displaystyle W in mathcal V y nbsp sho V V z displaystyle V in mathcal V z nbsp dlya vsih z W displaystyle z in W nbsp Todi z oznachennya U displaystyle U nbsp mayemo sho W U displaystyle W supset U nbsp i oskilki W V y displaystyle W in mathcal V y nbsp to z drugoyi vlastivosti takozh U V y displaystyle U in mathcal V y nbsp Oskilki tochka y displaystyle y nbsp bula dovilnoyu to U displaystyle U nbsp ye okolom vsih svoyih tochok tobto vidkritoyu mnozhinoyu u topologiyi t displaystyle tau nbsp dd Analogichno topologiyu mozhna zadavati za dopomogoyu bazi okoliv yak sim yu pidmnozhin B x x X displaystyle mathcal B x x in X nbsp sho zadovolnyayut vlastivosti yaki vikonuyutsya dlya baz okoliv Dlya kozhnogo x X displaystyle x in X nbsp B x displaystyle mathcal B x neq emptyset nbsp i dlya kozhnogo U B x displaystyle U in mathcal B x nbsp x U displaystyle x in U nbsp Yaksho U B x displaystyle U in mathcal B x nbsp to isnuye U V B x displaystyle U supset V in mathcal B x nbsp taka sho dlya kozhnoyi tochki y V displaystyle y in V nbsp isnuye U W B y displaystyle U supset W in mathcal B y nbsp Peretin skinchennoyi kilkosti elementiv B x displaystyle mathcal B x nbsp mistit deyakij element B x displaystyle mathcal B x nbsp Kardinalni funkciyi RedaguvatiZ ponyattyam bazi okoliv pov yazani nastupni ponyattya Harakter tochki x X displaystyle x in X nbsp u topologichnogo prostoru X displaystyle X nbsp najmensha mozhliva potuzhnist bazi okoliv u cij tochci Harakter tochki x X displaystyle x in X nbsp poznachayetsya x x X displaystyle chi x X nbsp Harakter prostoru X displaystyle X nbsp za oznachennyam rivnijx X sup x x X x X displaystyle chi X sup chi x X x in X nbsp Div takozh RedaguvatiBaza topologiyi Okil Topologichnij prostirDzherela RedaguvatiBurbaki N Zagalna topologiya Osnovni strukturi 3 e M Nauka 1968 S 276 Elementi matematiki ros Otrimano z https uk wikipedia org w index php title Sistema okoliv amp oldid 36724036
