About: Connected space

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Topological space that cannot be written as the disjoint union of two nonempty open subsets

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dbo:description
  • typ przestrzeni topologicznej (pl)
  • תכונה אפשרית של מרחב טופולוגי (iw)
  • množina, kterou nelze rozdělit na dvě disjunktní neprázdné otevřené podmnožiny (cs)
  • topological space that cannot be written as the disjoint union of two nonempty open subsets (en)
  • Spazio non rappresentato come l'unione di due o più insiemi aperti (it)
  • topologia spaco, kiu ne estas kunaĵo de du malfermitaj nemalplenaj subaroj, kies komunaĵo estas malplena (eo)
  • 2つ以上の互いに素な空でない開部分集合の和集合として表すことのできない位相空間 (ja)
  • topológiailag megszakítatlan része a térnek (hu)
  • Notion de topologie formalisant le fait pour un espace d'être d'un seul tenant (fr)
  • espacio topológico que no puede escribirse como la unión disjunta de dos subconjuntos abiertos no vacíos (es)
  • Beschreibung des Zusammenhangs eines topologischen Raumes in der Mathematik (de)
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  • right (en)
dbp:author
  • V. I. Malykhin (en)
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  • From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected spaces, whereas green space E is disconnected. Furthermore, A and B are also simply connected , while C and D are not: C has genus 1 and D has genus 4. (en)
dbp:direction
  • vertical (en)
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  • Connected and disconnected subspaces of R² (en)
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  • Simply connected, connected, and non-connected spaces.svg (en)
dbp:proof
  • By contradiction, suppose is not connected. So it can be written as the union of two disjoint open sets, e.g. . Because is connected, it must be entirely contained in one of these components, say , and thus is contained in . Now we know that: The two sets in the last union are disjoint and open in , so there is a separation of , contradicting the fact that is connected. (en)
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  • Proof (en)
  • Connected Set (en)
  • Connected space (en)
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  • ConnectedSet (en)
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  • 200 (xsd:integer)
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  • Connected space (en)
  • فضاء متصل (ar)
  • Conjunt connex (ca)
  • Souvislá množina (cs)
  • Koneksa spaco (eo)
  • Conjunto conexo (es)
  • Zusammenhängender Raum (de)
  • Ruang terhubung (in)
  • 連結空間 (ja)
  • Connexité (mathématiques) (fr)
  • Spazio connesso (it)
  • 연결 공간 (ko)
  • Przestrzeń spójna (pl)
  • Samenhang (nl)
  • Espaço conexo (pt)
  • Связное пространство (ru)
  • Зв'язний простір (uk)
  • Sammanhängande rum (sv)
  • 连通空间 (zh)
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