Uncountable Closed Ordinal Space is Lindelöf

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Theorem

Let $\Omega$ denote the first uncountable ordinal.

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


Then $\closedint 0 \Omega$ is a Lindelöf space.


Proof

We have:

Closed Ordinal Space is Compact
Compact Space is Lindelöf

$\blacksquare$


Sources