3 added 126 characters in body

EDIT: Both Georges's and Yemon's arguments are better since they avoid explanation why the group has to be metrizable.

No, there is no countably infinite compact group. The reason is that such group would be metrizable and hence a compact Polish space without isolated points. Into any such space one can embed the Cantor set, which is uncountable. The latter is not hard to prove, or you can look at Kechris, Classical Descriptive Set Theory, Theorem 6.2.

2 edited body

No, there is no countably infinite compact group. The reason is that such group would be metrizable and hence a compact Polish space without isolated points. Into any such space one can embed the Cantor set, which is uncountable. The letter latter is not hard to prove, or you can look at Kechris, Classical Descriptive Set Theory, Theorem 6.2.

1

No, there is no countably infinite compact group. The reason is that such group would be metrizable and hence a compact Polish space without isolated points. Into any such space one can embed the Cantor set, which is uncountable. The letter is not hard to prove, or you can look at Kechris, Classical Descriptive Set Theory, Theorem 6.2.