Proposition 4

There exists a unique topology on $\pi_1(B, a)$, compatible with its group structure, for which the admissible normal subgroups of $\pi_1(B, a)$ form a fundamental system of neighbourhoods of the identity element. For this topology, the open subgroups are exactly the admissible subgroups.

The following facts result from the definition of an admissible subgroup:

a) The group $\pi_1(B, a)$ is admissible;

b) A subgroup containing an admissible subgroup is admissible;

c) The intersection of a finite family of admissible subgroups is admissible;

d) For every admissible subgroup H of $\pi_1(B, a)$ the intersection of the subgroups $\gamma H\gamma^{-1}$, when $\gamma$ runs through $\pi_1(B, a)$, is admissible. In particular, the set of admissible normal subgroups is a filter basis formed of subgroups of $\pi_1(B, a)$ satisfying axiom (GV$'_{III}$) of III, p. 4, whence the first part of the proposition after TG, III, p. 5, example. The second part is then immediate (cf. TG, III, p. 7, corollary of prop. 4).

The topology on the group $\pi_1(B, a)$ characterized in proposition 4 is called the admissible topology.