Definition:Ordinal Space/Closed

Definition

Let $\Gamma$ be a limit ordinal.

The closed ordinal space on $\Gamma$ is the set $\closedint 0 \Gamma$ of all ordinal numbers less than or equal to $\Gamma$, together with the order topology.

Particular special cases of a closed ordinal space are as follows:

Closed Ordinal Space: $\Gamma < \Omega$

Let $\Omega$ denote the first uncountable ordinal.

The countable closed ordinal space on $\Gamma$ is a particular case of an closed ordinal space $\closedint 0 \Gamma$ where $\Gamma < \Omega$.

Closed Ordinal Space: $\Gamma = \Omega$

Let $\Omega$ denote the first uncountable ordinal.

The uncountable closed ordinal space on $\Omega$ is the particular case of a closed ordinal space $\closedint 0 \Gamma$ where $\Gamma = \Omega$.

Also see

• Definition:Open 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: $41, \ 43$. Closed Ordinal Space