Different profinite topologies on a group? 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.
 A: A good reference for both questions is Chapter 3 of Profinite groups by Ribes and Zalesskii.

Question 1
It is reasonable to say that there are many different "profinite topologies" on a group $G$. Section 3.1 of the above reference does the following. Let $G$ be a group and let $\mathcal{N}$ be a collection of finite-index normal subgroups of $G$, which is filtered from below, i.e., for any $N_1,N_2\in\mathcal{N}$ there is some $N\in\mathcal{N}$ such that $N\leq N_1\cap N_2$. Then one defines a topology on $G$ whose base is the collection of cosets of all elements of $\mathcal{N}$. Such a topology is called "a profinite topology" on $G$, and I'll call it the $\mathcal{N}$-profinite topology on $G$.
The special case when $\mathcal{N}$ is all (normal) finite-index subgroups of $G$ is usually referred to as "the" profinite topology on $G$. This terminology is justified by the following equivalent perspective. 
Let $\mathcal{C}$ be a collection of finite groups, which is closed under isomorphism, quotients, and finite subdirect products (Ribes and Zalesskii call this a formation of finite groups). Now let $G$ be any group, and let $\mathcal{N}$ be as before. If $\mathcal{N}\subseteq\mathcal{C}$ then the $\mathcal{N}$-profinite topology on $G$ is called "a pro-$\mathcal{C}$ topology on$G$". On the other hand, if $\mathcal{N}$ is precisely the collection of finite-index normal subgroups $N$ of $G$ such that $G/N$ is in $\mathcal{C}$, then the resulting topology is called "the pro-$\mathcal{C}$ topology".
If $\mathcal{C}$ is the class of all finite groups, then the pro-$\mathcal{C}$ topology is the the profinite topology. Other well-studied examples are when $\mathcal{C}$ is the class of finite $p$-groups, for some fixed prime $p$. In this case, the pro-$\mathcal{C}$ topology is also called the pro-$p$ topology. 

Question 2
Section 3.2 of the above reference deals with this.
Given a group $G$ and a collection $\mathcal{N}$ of finite-index normal subgroups filtered from below, one can construct the profinite completion of $G$ with respect to $\mathcal{N}$, which I will denote $G'_{\mathcal{N}}$. In particular, 
$$
G'_{\mathcal{N}}=\varprojlim_{N\in\mathcal{N}}G/N,
$$
which is a profinite topological group under the "sub-product" topology (i.e., the subspace topology induced by the product topology on $\prod_{N\in\mathcal{N}}G/N$). 
The special case when $\mathcal{N}$ is all normal finite-index subgroups of $G$ yields the usual profinite completion $G'$. 
Note that we have a canonical homomorphism $\tau_{\mathcal{N}}\colon G\to G'_{\mathcal{N}}$ which sends $g$ to $(gN)_{N\in\mathcal{N}}$. The connection to profinite topologies on $G$ is as follows. The $\mathcal{N}$-profinite topology on $G$  is precisely the coarsest topology on $G$ such that $\tau_{\mathcal{N}}$ is continuous  with respect to the sub-product topology on $G'_{\mathcal{N}}$. Dually, the sub-product topology on $G'_{\mathcal{N}}$ is precisely the finest topology on $G'_{\mathcal{N}}$ such that $\tau_{\mathcal{N}}$ is continuous with respect to the $\mathcal{N}$-profinite topology on $G$. (See  this link.) From this, one can compute relationships between bases for the two topologies, etc. 

Final Comment
Despite all of this, the terminology "profinite topology on $G$" is still possibly problematic for the following reason. Given any group $G$, the profinite completion $G'$ is a profinite topological group. On the other hand, while $G$ is a topological group under the profinite topology, it is not necessarily a profinite space (i.e., compact, Hausdorff, totally disconnected). For example, the profinite topology on $\mathbb{Z}$ is Hausdorff but not compact, since any countable compact Hausdorff group must be finite (see this question; the case of $\mathbb{Z}$ is also dealt with more explicitly here). The profinite topology on an infinite simple group is trivial, and so not Hausdorff or totally disconnected.  
Edit: To put a finer point on the previous comment, here is a quote from Profinite and residually finite groups by B. Hartley: "A topology on a group $G$ determined by a set $\mathcal{N}$ of normal subgroups of finite-index [satisfying certain properties] will be called a cofinite topology. Such topologies have been called profinite topologies by a number of authors; however we prefer to reserve that terminology for the situation when the resulting group is actually profinite."
