Let \(A\) be a \(k\)-affinoid algebra, a subset \(U\subset \mathrm{Spm}(A)\) is an affinoid subdomain if it is representable by a \(k\)-affinoid algebra \(A\xrightarrow{i} A'\). Let \(\mathrm{Hom}_{\mathrm{Spm}(\phi)\subset U}(A,)\) be the functor of affinoid morphisms whose image lies in \(U\), if \(A\to A'\) has the universal property
\[\mathrm{Hom}_{\mathrm{Spm}(\phi)\subset U}(A, B)\cong \mathrm{Hom}(A', B)\]
i.e. any \(k\)-affinoid morphism \(\phi: A\to B\) whose image \(\mathrm{Spm}(\phi)(\mathrm{Spm}(B))\subset \mathrm{Spm}(A)\) lies in \(U\), factors uniquely through \(A\xrightarrow{i} A'\to B\).
Yoneda’s lemma would tell us that the functor, or the set \(U\) uniquely determines the \(k\)-affinoid algebra \(A'\). It does not matter what the representation of \(A'\) is, they have an intrinsic geometric meaning: they are the functions living on \(U\).
Therefore for each affinoid subdomain \(U\subset \mathrm{Spm}(A)\), we can denote \(A_U\) the canonical \(k\)-affinoid algebra representing \(U\).
For \(U,U'\subset \mathrm{Spm}(A)\) affinoid subdomains, the intersection \(U\cap U'\) is also an affinoid subdomain, represented by the \(k\)-affinoid algebra \(A_U\widehat{\otimes}_A A_{U'}\).
The inverse image \(\mathrm{Spm}(\phi)^{-1}(U)\subset \mathrm{Spm}(B)\) is also an affinoid subdomain for any \(k\)-affinoid morphism \(\phi: A\to B\), this is obtained as the fibre product \(U\times_{\mathrm{Spm}(A)} \mathrm{Spm}(B)\), which is represented by the \(k\)-affinoid algebra \(A_U\widehat{\otimes}_A B\).
Theorem. Let \(A\) be a \(k\)-affinoid algebra, every affinoid subdomain \(U\subset \mathrm{Spm}(A)\) is a finite union of rational domains. \(\Rightarrow\) every affinoid subdomain is open in canonical topology.