Definition:Ordinal Space

Definition

The ordinal spaces are classified as follows:

Let $\Gamma$ be a limit ordinal.

Open Ordinal Space

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.

Closed Ordinal Space

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.

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: $40 \text { - } 43$. Ordinal Space