I have been reading the Berkeley lectures and got stuck with this question. Let $\mathbb{Q}_p [[t]]$ denote the ring of power series with $p$-adic coefficients. Is there a natural topology (e.g. the map from $\mathbb{Q}_p$ should be continuous) for which it's a Huber (also known as f-adic) ring?
I ask this because it seems that it can't have a topology with $\mathbb{Z}_p[[t]]$ being an open subring with the $(p,t)$-adic topology. Let us assume by way of contradiction this is the case, then $p\mathbb{Z}_p[[t]]$ is open since it is the preimage of $\mathbb{Z}_p[[t]]$ via the multiplication by $p^{-1}$ map. On the other hand, it is not open in the $(p,t)$-adic topology as for all $n$, $(p,t)^n$ does not lie in $(p)$. It is the case that $\operatorname{Spa}(\mathbb{Q}_p)\times \operatorname{Spa}(\mathbb{Z}[[t]])$ is $\bigcup_n\operatorname{Spa}(\mathbb{Q}_p\left<t,\frac{t^n}{p}\right>)$ (Berkeley lectures, page 30). It's an adic space that can be viewed as the next best thing to our question (so probably the best thing does not happen?).
Can you take $\mathbb{Z}_p\left<t\right>\subset \mathbb{Q}_p[[t]]$ as ring of definition with the $p$ adic topology? If so, what is the adic space it represents? Is there another natural option I missed? Also, I am still reading the basics so answers like "this is not a natural question since.." or "this ring is not suitible with this theory because blah blah blah" are very much welcome.
