Remark 5

Let B be a connected and locally arcwise connected topological space, let $a$ be a point of B, and let H be a subgroup of $\pi_1(B, a)$. By the preceding remark, if H is admissible, it is also open for the quotient topology of the topology of compact convergence. Conversely, suppose that H is a normal subgroup of $\pi_1(B, a)$ which is open for the quotient topology of the topology of compact convergence. Let $x$ be a point of B and let $\gamma \in \varpi_{a,x}(B)$. The subgroup $\gamma^{-1}H\gamma$ of $\pi_1(B, x)$ is again open for the quotient topology of the topology of compact convergence (III, p. 293, Remark 3). There therefore exists a neighbourhood V of $x$ in B such that $\gamma^{-1}H\gamma$ contains the class of every loop at $x$ whose image is contained in V. Thus $\gamma \pi_1(V, x)\gamma^{-1}\subset H$; since H is normal, this implies that H is an admissible subgroup of $\pi_1(B, a)$.