Exercise 2, § 5

Let X be a topological space and let $a$ be a point of X. A subgroup H of $\pi_1(X, a)$ is said to be adequate if, for every $b\in X$ and every path class $\gamma \in \varpi_{a,b}(X)$, there exists a neighbourhood V of $b$ such that for every loop $c\in \Omega_b(V)$, the class of the loop $\gamma [c]\gamma^{-1}$ belongs to H.

a) Show that there exists a unique group topology on $\pi_1(X, a)$ for which the adequate subgroups are the open subgroups.

b) Show that this topology is finer than the compact convergence topology but that these two topologies have the same distinguished open subgroups.

c) If X is locally connected by arcs, the adequate topology and the compact convergence topology have the same open subgroups.