I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
Definition: Suppose the group $G$ contains a family $\Theta(V_i)$ of subgroups of $G$ such that:
(1) $\bigcap_i V_i = 1$ all $V_i \in \Theta$
(2) If $V,V' \in \Theta$ then $V\cap V' \in \Theta$
(3) If $V \in \Theta$ then there is a normal subgroup $V' \in \Theta$ such that $V' \subseteq V$.
If we take all sets $xV, Vx$, $V \in \Theta, x \in G$ as a basis for open sets in $G$, then $G$ becomes a topological group. We say that this is a subgroup topology for $G$ defined by the family $\Theta$.
At two points he makes references to Andre Weil and his "voisinage's" and on p. 131 in particular:
Following A. Weil [10, page 10] a group $G$ with a subgroup topology defined by a family $\Theta(V_i)$ of subgroups is a uniform space. We suppose the subgroups $V_i$ indexed. Here for $x \in G$, $V_i(x)$ is the coset $V_ix$. Thus in the space $G\times G$ we define $V^i$ to be the set of pairs $(p,q)$ where $q \in V_i(p)$. In terms of this uniform structure [10, pp. 17-19] $G$ my be completed by a Cauchy sequence to a complete space $\hat{G}$.
This paragraph I cannot follow, first of all I do not have access to Weil's publication (and even if I would I can not read french). So are there any other resources where I can find more about the constructions done here, in particular what is meant by Cauchy sequence, I just know them for metric spaces?
