Theorem 63.27.2 (Grothendieck). Let $X$ be a connected scheme.

1. There is a topology on $\pi _1(X, \overline{x})$ such that the open subgroups form a fundamental system of open nbhds of $e\in \pi _1(X, \overline x)$.

2. With topology of (1) the group $\pi _1(X, \overline{x})$ is a profinite group.

3. The functor

$\begin{matrix} \text{ schemes finite } \atop \text{ étale over }X & \to & \text{ finite discrete continuous } \atop \pi _1(X, \overline{x})\text{-sets} \\ Y / X & \mapsto & F_{\overline{x}}(Y) \text{ with its natural action} \end{matrix}$

is an equivalence of categories.

Proof. See [SGA1]. $\square$

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