Definition:Ordinal Space/Open/Uncountable

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Let $\Gamma$ be a limit ordinal.

Let $\Omega$ denote the first uncountable ordinal.

The uncountable open ordinal space on $\Omega$ is the particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma = \Omega$.

That is, it is the set $\hointr 0 \Omega$ of all ordinal numbers (strictly) less than $\Omega$, together with the order topology.

Also see

  • Results about ordinal spaces can be found here.