Remark 4

The admissible topology on $\pi_1(B, b)$ is the least fine topology for which the action of $\pi_1(B, b)$ on $E_b$ is continuous for every connected covering E of B (Prop. 6 of III, p. 318 and Prop. 7 of III, p. 318). If E is a connected covering of B and $x$ a point of the fiber $E_b$, the mapping $c'\mapsto c'(1)$ of $\Lambda_x(E)$ into E is continuous. Consequently, the mapping $c\mapsto x\cdot c$ of $\Lambda_b(B)$ into E is continuous (III, p. 302, Cor. 2 of Prop. 3). By restriction to $\Omega_b(B)$ and passing to the quotient, it follows that the mapping $\gamma \mapsto x\cdot \gamma$ is continuous when the group $\pi_1(B, b)$ is endowed with the quotient-space topology of $\Omega_b(B)$. The admissible topology on $\pi_1(B, b)$ is therefore less fine than the quotient topology of the topology of compact convergence. It may be strictly less fine (III, p. 337, Exercise 7).