Lemma 57.14.2. Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme over $k$. Then $X_{k^{perf}}$ is connected and $\pi _1(X_{k^{perf}}) \to \pi _1(X)$ is an isomorphism.

Proof. Special case of topological invariance of the fundamental group. See Proposition 57.8.4. To see that $\mathop{\mathrm{Spec}}(k^{perf}) \to \mathop{\mathrm{Spec}}(k)$ is a universal homeomorphism you can use Algebra, Lemma 10.45.10. $\square$

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