Proposition 6

Let B be a topological space and let $b$ be a point of B. Endow the group $\pi_1(B, b)$ with the admissible topology.

a) For every discrete group G, every principal covering E of B with group G, and every $x\in E_b$, the homomorphism $h_{(E,x)}$ is a continuous homomorphism of topological groups.

b) For every covering E of B, the canonical action of $\pi_1(B, b)$ on the fibre $E_b$ is admissible.

It is enough to prove the second assertion. Let K be the set of elements $k\in \pi_1(B, b)$ such that $x\cdot k=x$ for every $x\in E_b$; let us prove that K is an admissible subgroup of $\pi_1(B, b)$.

Let $b'$ be a point of B and let $V'$ be a neighbourhood of $b'$ in B over which the covering E is trivializable. Let $\gamma$ be the class of a path $c$ joining $b$ to $b'$ in B and let $c'$ be the unique path in E with origin $x$ which lifts $c$. For every $\delta \in \pi_1(V', b')$ and every $x'\in E_{b'}$, one has $x'\cdot \delta =x'$. Consequently, for $x\in E_b$ and $\delta \in \pi_1(V', b')$, one has

so that $\gamma \delta \gamma^{-1}$ belongs to K. In other words, K is an admissible subgroup, whence the proposition.