The Stacks project

Theorem 64.27.2 (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

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

    is an equivalence of categories.

Proof. See [SGA1]. $\square$

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