Definition:Ordinal Space/Closed/Uncountable

Definition

Let $\Gamma$ be a limit ordinal.

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 $\Gamma = \Omega$.

That is, it is the set $\closedint 0 \Omega$ of all ordinal numbers less than or equal to $\Omega$, together with the order topology.

Also see

• Definition:Uncountable Open Ordinal Space

• Definition:Countable Closed Ordinal Space
• Definition:Countable 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: $43$. Closed Ordinal Space $[0, \Omega]$