Is every countably generated profinite group countably based?

Question

In a profinite group: Does the existence of a countable generating (topologically) set imply the existence of a countable basis for the topology.

Answer 1

This depends on what you mean by countably generated. If you mean by a countable set of generators converging to 1 then yes, see the Ribes and Zalesskii book.

If you mean a countable set generating a dense subgroup the answer is no. If you take the free profinite group on a countably infinite discrete space X (not in the converging to 1 sense) then the closure of X is the Stone-Čech compactification of X which is not countably based.