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E33

[0HR] Prerequisites:[0KX],[0KZ].Let X=ℝβˆͺ{∞}, let’s consider the family B of parts of X comprised of

  • the open intervals (a,b) with a,bβˆˆβ„ and a<b,

  • the sets (a,+∞)βˆͺ(βˆ’βˆž,b)βˆͺ{∞} with a,bβˆˆβ„ and a<b.

Show that B satisfies the properties (a),(b) seen in [0KX]. Let 𝜏 therefore be the topology generated by this base. The topological space (X,𝜏) is called one-point compactified line. This topological space is T2 and it is compact (Exer. [0JD]); it is homeomorphic to the circle (Exer. [0YF]); therefore it can be equipped with a distance that generates the topology described above.

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Bibliography
Book index
  • space, topological
  • topological space
  • base, (topology)
  • one-point compactified line
  • real line, one-point compactified ---
  • real line , see also real numbers
  • real numbers , see also real line
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