Definition:Ordinal Space/Open/Countable
< Definition:Ordinal Space | Open(Redirected from Definition:Countable Open Ordinal Space)
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Definition
Let $\Gamma$ be a limit ordinal.
Let $\Omega$ denote the first uncountable ordinal.
The countable open ordinal space on $\Gamma$ is a particular case of an open ordinal space $\hointr 0 \Gamma$ where $\Gamma < \Omega$.
That is, it is the set $\hointr 0 \Gamma$ of all ordinal numbers (strictly) less than $\Gamma < \Omega$, together with the order topology.
Also see
- Results about ordinal spaces can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $40$. Open Ordinal Space $[0, \Gamma) \ (\Gamma < \Omega)$