Basis for Open Ordinal Topology
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This article, or a section of it, needs explaining, namely:
It's unclear from the source work used whether this is the basis for the Open Ordinal Topology or Closed Ordinal Topology.
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This theorem requires a proof..
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Theorem
Let $\Gamma$ be a limit ordinal.
Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.
Consider the set $\BB$ of subsets of $\hointr 0 \Gamma$ of the form:
- $\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$
for $\alpha, \beta \in \hointr 0 \Gamma$.
It's unclear from the source work used whether this is the basis for the Open Ordinal Topology or Closed Ordinal Topology.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Then $\BB$ forms a basis for $\hointr 0 \Gamma$.
Proof
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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
