I have some general questions around the profinite topology on a group $G$.
On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that
The profinite topology on a group is a topology on the underlying set of the group defined in the following equivalent ways:
- It has as a basis of open subsets all left cosets of subgroups of finite index.
- It has as a basis of open subsets all right cosets of subgroups of finite index.
- It has as a basis of open subsets all cosets of normal subgroups of finite index.
Under the profinite topology, any group becomes a topological group.
I always hear, that there is ONE profinite topology on a group. But every set of subgroups of finite index, which is closed under taking intersections gives us a "profinite" topology for a group. But then maybe some elements of the above basiselements aren't open sets in this topology. So are these topologies also called profinite, although they aren't the above defined one?
If we have a profinite group. How is the connection between the "profinite topology" and the definition of a profinite group. If we got a "profinite" topology on $G$. What about the profinite completion $G'$. How can I find a basis of open subsets in $G'$. How is the connection between a basis of $G$ and a basis of $G'$?
Thanks for help.