Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.

Consider the sets $U := \{ |T_1| < 1\}$ and $V := \{ |T_2| = 1\}$. It's standard that $U$ and $V$ are both admissible open subsets of $X$ (for the strong $G$-topology); e.g. this follows from Propositions 9.1.4/4 and 9.1.4/5 of Bosch--Guentzer--Remmert. However, it's not part of the axioms for a G-topology that the union of two admissible open sets be admissible open.

• Is $U \cup V$ an admissible open set for the $G$-topology of $X$?
• If so, is $\{ U, V\}$ an admissible covering of $U \cup V$?