Khipha izakhiwo ezimbili ezanele ukukhombisa ukuthi isethi engezansi yesikhala sika-Euclidean sihlangene. Beka lezi zakhiwo ohlwini olunezinhlamvu.
Context:Kungase kusetshenziswe izincazelo ezihlukahlukene zokuqina, kuye ngezinga lokuvamile. Iqoqo elincane lesikhala sika-Euclide ngokukhethekile libizwa ngokuthi i-compact uma livaliwe futhi livinjelwe. Lokhu kusho, ngeBolzano-Weierstrass theorem, ukuthi noma yikuphi ukulandelana okungenamkhawulo kusuka kusethi kunokulandelana okuhambisana nephuzu kusethi. Imibono ehlukahlukene elinganayo yokuqina, njengokuqina okulandelanayo nokuqina kwephuzu lomkhawulo, kungathuthukiswa ezindaweni ezijwayelekile zemethrikhi.  Ngokuphambene nalokho, imibono ehlukene yokuqina ayifani ezindaweni ezijwayelekile ze-topological, futhi umqondo owusizo kakhulu wokuqina <unk> ekuqaleni obizwa ngokuthi yi-bicompactness <unk> uchazwa ngokusebenzisa izembozo eziqukethe amasethi avulekile (bheka Incazelo yesembozo esivulekile ngezansi). Ukuthi lolu hlobo lokuhlanganiswa lubambelela kuma-subset avaliwe futhi aboshwe wendawo ye-Euclidean yaziwa ngokuthi i-Heine<unk>Borel theorem. Ukuqina, lapho kuchazwa ngale ndlela, kuvame ukuvumela umuntu ukuba athathe ulwazi olwaziwa endaweni <unk> endaweni eseduze nendawo ngayinye <unk> futhi alwandise lube ulwazi oluhambisana nomhlaba wonke kuyo yonke indawo. Isibonelo salokhu kuyinto Dirichlet sika theorem, okuyinto ekuqaleni isicelo Heine, ukuthi umsebenzi okuqhubekayo ku isikhawu compact kuyinto ngokulinganayo okuqhubekayo; lapha, okuqhubekayo kuyinto impahla wendawo umsebenzi, futhi okuqhubekayo iyunifomu elihambisana impahla global.
Ngokwe-theorem ka-Heine-Borel lezi zakhiwo ezilandelayo zanele ukukhombisa ukuthi isethi engaphansi yesikhala se-Euclidean sihlangene: - Isethi ivaliwe - Isethi ibekelwe imingcele