Definition:Ordinal Space/Open

Definition

Let $\Gamma$ be a limit ordinal.

The open ordinal space on $\Gamma$ is the set $\hointr 0 \Gamma$ of all ordinal numbers (strictly) less than $\Gamma$, together with the order topology.

Particular special cases of an open ordinal space are as follows:

Open Ordinal Space: $\Gamma < \Omega$

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 $\Gamma < \Omega$.

Open Ordinal Space: $\Gamma = \Omega$

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 $\Gamma = \Omega$.

Also see

• Definition: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: $40, \ 42$. Open Ordinal Space