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What can be said about the algebra of continuous functions on compact countable ordinals?
Let $X$ be a compact countable Hausdorff space. By Sierpinski-Mazurkiewicz Theorem we know that $X$ is a compact countable ordinal, i.e.
$$
X \simeq \omega ^{\alpha} \cdot n + 1
$$
where $\alpha$ is ...
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