Bvisa zvinhu zviviri zvinokwana kuratidza kuti chikamu chenzvimbo yeEuclidean chakamanikana. Taura zvinhu zvacho zviri muzvinyorwa zvine mavara.
Context:Tsananguro dzakasiyana-siyana dzokuoma dzingagona kushanda, zvichienderana nechiyero chechimiro. Chikamu chiduku chenzvimbo yeEuclidean zvikurukuru chinonzi chakasimba kana chakavharwa uye chakaganhurirwa. Izvi zvinoreva, neBolzano-Weierstrass theorem, kuti chero nhevedzano isingaperi kubva muunganidzwa ine nhevedzano inotevera inosangana nepfungwa iri muunganidzwa. Pfungwa dzakasiyana-siyana dzakaenzana dzokuoma, dzakadai sokuoma kwakatevedzana uye kuoma kwakaperera, dzinogona kugadzirwa munzvimbo dzomumetric.  Mukupesana, pfungwa dzakasiyana dzekushomeka hadzina kuenzana munzvimbo dzese dzepamusoro, uye pfungwa inonyanya kubatsira yekushomeka <unk> pakutanga inonzi bicompactness <unk> inotsanangurwa uchishandisa mabutiro anosanganisira akavhurika seti (ona yakavhurika butiro tsananguro pazasi). Kuti rudzi urwu rwekupfupika runobata nokuda kwezvikamu zviduku zvakavharwa uye zvakabatanidzwa zveEuclidean nzvimbo inozivikanwa seHeine<unk>Borel theorem. Kukwana, apo kunorondedzerwa neiyi nzira, kazhinji kunobvumira mumwe kutora mashoko anozivikanwa munharaunda <unk> munharaunda yenzvimbo imwe neimwe yenzvimbo <unk> uye kuikudza kumashoko anobata munyika yose munzvimbo yose. Muenzaniso wechiitiko ichi ndiDirichlet's theorem, iyo yakatanga kushandiswa naHeine, kuti basa risingagumi pane imwe nguva yakaoma yakafanana; pano, kuenderera mberi kune nzvimbo yebasa, uye kuenderera mberi kwakaringana iyo inoenderana nenyika yose.
Maererano ne theorem yaHeine-Borel zvinhu zvinotevera zvinokwana kuratidza kuti subset yeEuclidean space yakaoma: - Set yakavharwa - Chikwata chacho chakaganhurirwa