Ordinal Space

See: Ordinal, Space, Ordinal Number, Order Topology.

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References

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Order_topology#Ordinal_space
  • For any ordinal number λ one can consider the spaces of ordinal numbers
    • [0,\lambda) = \{\alpha \mid \alpha < \lambda\}\,
    • [0,\lambda] = \{\alpha \mid \alpha \le \lambda\}\,
  • together with the natural order topology. These spaces are called ordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0,λ) and λ + 1 = [0,λ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the discrete topology.
  • When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual (still discrete) topology, while [0,ω] is the one-point compactification of N.
  • Of particular interest is the case when λ = ω1, the set of all countable ordinals, and the first uncountable ordinal.

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