V matematici zokrema v zagalnij topologiyi evklidova abo prirodna topologiya ye odniyeyu z topologij zadanih na mnozhini vsih dijsnih chisel R displaystyle mathbb R Yiyi standartnu bazu skladayut intervali a b x R a lt x lt b displaystyle a b x in mathbb R mid a lt x lt b a b R displaystyle a b in mathbb R a lt b displaystyle a lt b 1 Vlastivosti red Evklidova topologiya na R displaystyle mathbb R nbsp porodzhena evklidovoyu metrikoyu d x y x y displaystyle d x y x y nbsp na R displaystyle mathbb R nbsp de x displaystyle x nbsp oznachaye absolyutne znachennya modul dijsnogo chisla x displaystyle x nbsp Takim chinom metrichnij prostir R displaystyle mathbb R nbsp zadovolnyaye vsi aksiomi vidokremlyuvanosti Krim togo R displaystyle mathbb R nbsp ye povnim metrichnim prostorom drugoyi kategoriyi R displaystyle mathbb R nbsp zadovolnyaye drugu aksiomu zlichennosti oskilki mnozhini viglyadu a b displaystyle a b nbsp de a displaystyle a nbsp i b displaystyle b nbsp racionalni ye zlichennoyu bazoyu R displaystyle mathbb R nbsp Tomu R displaystyle mathbb R nbsp zadovolnyaye pershu aksiomu zlichennosti ye lindelofovim i separabelnim Mnozhina racionalnih chisel ye zlichennoyu skriz shilnoyu v R displaystyle mathbb R nbsp mnozhinoyu R displaystyle mathbb R nbsp ne ye zlichenno kompaktnim prostorom bo vidkriti intervali n n 2 displaystyle n n 2 nbsp dlya vsih cilih n pokrivayut R displaystyle mathbb R nbsp ale zhodna yih skinchenna sukupnist ne ye pokrittyam R displaystyle mathbb R nbsp Ale R displaystyle mathbb R nbsp lokalno kompaktnij i s kompaktnij oskilki vidrizki a b displaystyle a b nbsp a b R displaystyle a b in mathbb R nbsp a lt b displaystyle a lt b nbsp kompaktni Bud yaka zamknena v R displaystyle mathbb R nbsp mnozhina A displaystyle A nbsp ye G s displaystyle G sigma nbsp mnozhinoyu oskilki A n 1 A n displaystyle A bigcap n 1 infty A n nbsp de A n displaystyle A n nbsp okil mnozhini A displaystyle A nbsp radiusu 1 n displaystyle 1 n nbsp n N displaystyle n in mathbb N nbsp tobto A n x A B x 1 n displaystyle A n bigcup x in A B x 1 n nbsp Kozhna tochka sho ne nalezhit A displaystyle A nbsp mistitsya v e okoli yakij ne peretinayetsya z A displaystyle A nbsp i takim chinom ne peretinayetsya z deyakim A n displaystyle A n nbsp Bud yake vidkrite pokrittya R displaystyle mathbb R nbsp pokrivaye kozhen kompaktnij vidrizok n n 1 displaystyle n n 1 nbsp n Z displaystyle n in mathbb Z nbsp tomu vidkrite pokrittya mozhe buti zmenshene do poslidovnosti skinchennih pidpokrittiv G i n displaystyle G i n nbsp kozhnogo vidrizka n n 1 displaystyle n n 1 nbsp Todi mnozhini G i n n 1 n 2 displaystyle G i n cap n 1 n 2 nbsp utvoryuyut lokalno skinchenne pokrittya vpisane v pochatkove vidkrite pokrittya Takim chinom R displaystyle mathbb R nbsp parakompaktnij Topologiya na R displaystyle mathbb R nbsp takozh mozhe buti zadana kvazimetrikoyu d x y y x displaystyle d x y y x nbsp koli y x displaystyle y geqslant x nbsp i d x y 2 x y displaystyle d x y 2 x y nbsp koli y lt x displaystyle y lt x nbsp Nabir mnozhin S a b x y x y lt b displaystyle S ab x y x y lt b nbsp chi x y gt a displaystyle x y gt a nbsp de a b R displaystyle a b in mathbb R nbsp i a lt b displaystyle a lt b nbsp ye peredbazoyu rivnomirnosti U displaystyle U nbsp porodzhenoyi prirodnoyu topologiyeyu na R displaystyle mathbb R nbsp ale U displaystyle U nbsp ne ye zvichajnoyu metrichnoyu rivnomirnistyu Evklidiv n displaystyle n nbsp vimirnij prostir R n displaystyle mathbb R n nbsp viznachayetsya yak dobutok n kopij R displaystyle mathbb R nbsp Topologiya dobutku porodzhuyetsya bazoyu yaka skladayetsya z vidkritih pryamokutnikiv tobto mnozhin yaki ye dekartovim dobutkom vidkritih intervaliv z kozhnoyi kopiyi R displaystyle mathbb R nbsp Ekvivalentna baza skladayetsya z vidkritih n displaystyle n nbsp vimirnih kul vidnosno evklidovoyi metriki d x y S x i y i 2 1 2 displaystyle d x y Sigma x i y i 2 frac 1 2 nbsp v R n displaystyle mathbb R n nbsp Literatura red Steen Lynn Arthur Seebach J Arthur Jr 1995 1978 Counterexamples in Topology vid Dover reprint of 1978 Berlin New York Springer Verlag ISBN 978 0 486 68735 3 MR 507446 Otrimano z https uk wikipedia org w index php title Evklidova topologiya dijsnoyi pryamoyi amp oldid 38824607
