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2

(Not really an answer, but too long for a comment, bear with me.) I'd say that those conditions will be pretty subtle and I doubt anything both universal and meaninful can be said. For example, let's restrict attention to sphere bundles which come as unit sphere bundles of vector ones of rank $r$, so we have fibration $S^{r-1} \to E \to B$. If cohomology ...

0

Peter Scholze's comment gives a good answer to the main direct question and tells us that the condensed mod p vector space with basis a compact Hausdorff space S is solid if and only if S is finite. The advantages of solidification are going to become more apparent as we use the condensed maths more widely. Solidification is particularly important when using ...

10

Here's a sketch of the proof. I encourage you to fill in the details yourself. The definition of $V_K$ is $V_K=H^0(G_{\overline K/K},V)$. The key part of the proof is to show that $V$ has a $\overline{K}$ basis consisting of vectors in $V_K$. To find such a basis, start with an arbitrary basis $v_1,\ldots,v_n\in V$. Then each $\sigma\in G_{\overline K/K}$ ...

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