I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties. 


In the literature (e.g. in Coleman-Mazur) I've seen people talk about the $\mathbf{C}\_p$-points of a rigid analytic space over $\mathbf{Q}\_p$ (here $\mathbf{C}\_p$ is the completion of an algebraic closure of $\mathbf{Q}\_p$). Based on the case of schemes, if $X$ and $T$ are rigid spaces over $\mathbf{Q}\_p$, I assume the set of $T$-valued points of $X$ is the set $\mathrm{Hom}\_{\mathbf{Q}\_p}(T,X)$, where the morphisms are in the category of $G$-topologized spaces over $\mathrm{Sp}(\mathbf{Q}\_p)$. But $\mathbf{C}\_p$ is not a $\mathbf{Q}\_p$-affinoid algebra, so, strictly speaking, it's associated $G$-topologized space $\mathrm{Sp}(\mathbf{C}\_p)$ is not a rigid space over $\mathbf{Q}\_p$, right? I guess there is a morphism of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}\_p)\rightarrow\mathrm{Sp}(\mathbf{Q}\_p)$, and so one could define the $\mathbf{C}\_p$-points of $X$ in the same way as I did above, just morphisms of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}\_p)\rightarrow X$ compatible with the morphisms to $\mathrm{Sp}(\mathbf{Q}\_p)$. 


Question: Is this what is meant by the $\mathbf{C}\_p$-points of a rigid analytic space over $\mathbf{Q}\_p$?


The only alternative I can think of is to define the $\mathbf{C}\_p$-points of a rigid space $X$ over $\mathbf{Q}\_p$ as the $\mathbf{C}\_p$-valued points of the base change $X\_{\mathbf{C}\_p}$. Although, if my intuition drawn from schemes is to be trusted, this is probably equivalent to the definition I suggested above.



How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?
I think in Coleman-Mazur, it is just to be taken in the sense of $\mathbb C\_p$-points of $X\_{\mathbb C\_p}$. This is compatible with the more abstract definition you propose. Indeed, as $\mathrm{Sp}(\mathbb C\_p)$ is just one point, it is enough to work with an affinoide space $X$ associated to an affinoide algebra $A$ over $\mathbb Q\_p$. Then it is clear that the continuous homomorphisms from $A$ to $\mathbb C\_p$ coincide with continuous homomorphisms from $A\widehat{\otimes}\_{\mathbb Q\_p} \mathbb C\_p$ to $\mathbb C\_p$.


