Definition:Ordinal Space/Open/Uncountable

Definition

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

• Definition:Countable Open Ordinal Space
• Definition:Countable Closed Ordinal Space

• Definition:Uncountable Closed Ordinal Space

• 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: $42$. Open Ordinal Space $[0, \Omega)$