Definition:Ordinal Space/Closed
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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
- 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