Exercise 1, § 5

Let $X$ be a space connected by arcs, let $x$ be a point of $x$, and let $\mathscr{U}= (U_i)$ be an open covering of X; for every $i$, let $c_i$ be the class of a path in X having origin a point $u_i$ of $U_i$ and end $x$. Denote by $G_{\mathscr{U}}$ the smallest normal subgroup of $\pi_1(X, x)$ such that Int($c_i$)$(G_{\mathscr{U}})$ contains the class of every loop in $U_i$ at $u_i$. Then $G_{\mathscr{U}}$ is an open subgroup of $\pi_1(X, x)$ for the admissible topology.