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Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.

Is there a compact (Hausdorff, or even T1) (topological) group which is infinite, but has countable cardinality? The "obvious" choices don't work; for instance, \mathbb{Q}/\mathbb{Z} (with the obvious induced topology) is non-compact, and I get the impression that profinite groups are all uncountable (although I might be wrong there). So does someone have an example, or a reference in the case that there are no such groups?
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Is there a compact group of countably infinite cardinality?

Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.

Is there a compact (topological) group which is infinite, but has countable cardinality? The "obvious" choices don't work; for instance, \mathbb{Q}/\mathbb{Z} (with the obvious induced topology) is non-compact, and I get the impression that profinite groups are all uncountable (although I might be wrong there). So does someone have an example, or a reference in the case that there are no such groups?