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Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho \mathbb{R}}$, $\mathbb{P}^{1,an}_{^\rho \mathbb{R}}$ (or, if this is easier, the same with the algebraic closure $\mathbb{A}^{1,an}_{^\rho \mathbb{C}}$, $\mathbb{P}^{1,an}_{^\rho \mathbb{C}}$) ? (this might be a $\aleph_\alpha$-valued $\mathbb{R}$-tree, but I don't know this for sure)

I know this might not be the way to go about non-standard analysis, but I wonder if there is a connection between Berkovich spaces and non-standard spaces in this way.

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