The de Jong fundamental group of ${\mathbb{P}}_C^1$ depends on $C$ and is not always topologically countably generated

Abstract

For $C/{\mathbb{Q}}_p$ complete and algebraically closed, we show that the de Jong fundamental group ${\pi }_{1,\mathrm{dJ}}({\mathbb{P}}_C^1)$ depends on $C$ and, if $C$ has cardinality $>2^{\mathbb{N}}$, that it is not topologically countably generated. The result and proofs generalize to any connected rigid analytic variety with a non-constant map to ${\mathbb{P}}_C^n$.

1 page (+ references) v3 - added a second proof