Omega as Limit Point of Intervals of Uncountable Closed Ordinal Space

Theorem

Let $\Omega$ denote the first uncountable ordinal.

Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.

Then $\Omega$ is a limit point of the set $\openint a \Omega$, but not the limit of any sequence of points in $\openint a \Omega$.

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: $43$. Closed Ordinal Space $[0, \Omega]$: $2$