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?



Is there a compact group of countably infinite cardinality?
No, there is no countably infinite compact Hausdorff topological group.


Indeed such a group $G$ would have a left-invariant Haar measure $m$ with $m(G)=1$
and all points would have the same measure (since the group acts transitively on itself).
But then, by countable additivity of the measure $m$, the group itself would have measure $m(G)=0$ or $m(G)=\infty$ according as its points all had $m(p)=0$ or $m(p)>0$ . A contradiction in both cases to the fact that $m(G)=1$ .


