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English: Examples and non-examples of topological spaces, based roughly on Figures 12.1 and 12.2 from Munkres' Introduction to Topology. The 6 examples are subsets of the power set of {1,2,3}, with the small circle in the upper left of each denoting the empty set, and in reading order they are:
  • {{}, {1,2,3}} (the trivial topology)
  • {{}, {1}, {1,2,3}}
  • {{}, {1}, {2}, {1,2}, {1,2,3}}
  • {{}, {2}, {1,2}, {2,3}, {1,2,3}}
  • {{}, {2}, {3}, {1,2,3}} (not a topology, because {2} and {3} are in the topology but their union {2,3} is not)
  • {{}, {1,2}, {2,3}, {1,2,3}} (not a topology, because {1,2} and {2,3} are in the topology but their intersection {2} is not)
Deutsch: Beispiele und Gegenspiele zu topologischen Räumen, die grob den Abbildungen 12.1 und 12.2 aus dem Buch Introduction to Topology von Munkres entsprechen. Die sech Beispiele sind Teilmenge der Potenzmenge von {1,2,3}. Der kleine Kreis in den Beispielen oben links bezeichnet die leere Menge. Im Detail lauten die Beispiele dann:
  • {{}, {1,2,3}} (triviale Topologie)
  • {{}, {1}, {1,2,3}}
  • {{}, {1}, {2}, {1,2}, {1,2,3}}
  • {{}, {2}, {1,2}, {2,3}, {1,2,3}}
  • {{}, {2}, {3}, {1,2,3}} (kein topologischer Raum, da die Mengen {2} und {3} zur Topologie gehören, aber ihre Vereinigung {2,3} nicht)
  • {{}, {1,2}, {2,3}, {1,2,3}} (kein topologischer Raum, da die Mengen {1,2} und {2,3} zur Topologie gehören, aber der Schnitt {2} der beiden nicht)
Español: Ejemplos y contraejemplos de espacios topológicos basados, fundamentalmente, en las Figuras 12.1 y 12.2 del libro "Introduction to Topology" de Munkres. Los seis ejemplos son subconjuntos del conjunto potencia {1,2,3}, donde el círculo pequeño en la esquina superior izquierda de cada figura representa el conjunto vacío, y que en orden de lectura son
  • {{}, {1,2,3}} (topología trivial)
  • {{}, {1}, {1,2,3}}
  • {{}, {1}, {2}, {1,2}, {1,2,3}}
  • {{}, {2}, {1,2}, {2,3}, {1,2,3}}
  • {{}, {2}, {3}, {1,2,3}} (no es topológico porque {2} y {3} se encuentran en la topología, pero su unión {2,3} no)
  • {{}, {1,2}, {2,3}, {1,2,3}} (no es topológico porque {1,2} y {2,3} se encuentran en la topología, pero su intersección {2} no)
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current06:54, 26 May 2008Thumbnail for version as of 06:54, 26 May 2008360 × 319 (12 KB)Dcoetzee{{Information |Description=Examples and non-examples of topological spaces, based roughly on Figures 12.1 and 12.2 from Munkres' ''Introduction to Topology''. The 6 examples are subsets of the power set of {1,2,3}, with the small circle in the upper left

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