Consider the following generalization of topological spaces:

**Definition:** Let $X$ be a set. A *G-topology* on $X$ is given by certain distinguished subsets $U \subset X$, called *admissible open subsets*, as well as for each admissible open subset $U \subset X$ a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called *admissible coverings*, in such a way that the following axioms are satisfied:

- The intersection $U \cap V$ of two admissible open subsets $U, V \subset X$ is again admissible open.
- For each admissible open subset $U \subset X$, the trivial covering $\{U\}$ of $U$ is admissible.
- If $U$ is an admissible open subset of $X$, $\{U_i\}_{i \in I}$ is an admissible covering of $U$ and for each $i \in I$ the family $\{V_{ij}\}_{j \in J_i}$ is an admissible covering of $U_i$, then the family $\{V_{ij}\}_{i \in I, j \in J_i}$ is an admissible covering of $U$.
- If $U, V \subset X$ are two admissible open subsets of $X$ and $V \subset U$, and if $\{U_i\}_{i \in I}$ is an admissible covering of $U$, then the family $\{V \cap U_i\}_{i \in I}$ is an admissible covering of $V$.

The admissible open subsets, ordered by inclusion, form a partially ordered set and hence a category. The covering data gives rise in an obvious way to a Grothendieck pre-topology and hence to a Grothendieck topos $\mathrm{Sh}(X)$ of sheaves on $X$. G-topological spaces appear prominently in rigid analytic geometry. The basic theory of G-topological spaces and sheaves on them is developed for instance in [BGR84, Sec. 9.1 and Sec. 9.2].

I know from [MLM92, Thm. 1 in Sec. IX.5] that a site whose underlying category is a poset gives rise to a localic topos, i.e. we must have $\mathrm{Sh}(X) = \mathrm{Sh}(X_{\mathrm{loc}})$ for some (uniquely determined) locale $X_{\mathrm{loc}}$. I also know that $X_{\mathrm{loc}}$ can be described as the locale of subobjects in the terminal object of $\mathrm{Sh}(X)$. However I wonder:

**Questions:** Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I think that this is not in general the case.)
Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

**Edit:** Here is another remark: Any admissible open subset $U \hookrightarrow X$ gives rise to a morphism of topoi $\mathrm{Sh}(U) \to \mathrm{Sh}(X)$ and hence (by [MLM92, Prop. 2 in Sec. IX.5]) to a morphism $U_{\mathrm{loc}} \to X_{\mathrm{loc}}$. I think that by [MLM92, Prop. 5 (ii) in Sec. IX.5] this map is an embedding of locales, so both the ordered set of admissible open subsets of $X$ and the frame of opens of $X_{\mathrm{loc}}$ should embed in the frame of sublocales of $X_{\mathrm{loc}}$. Can we describe the relation between the two within this bigger poset?

For a concrete example, consider the set of real numbers $\mathbb{R}$ an consider the G-topological space with underlying set $\mathbb{R}$ where

- a subset $I \subset \mathbb{R}$ is admissible open if it is a compact interval, and
- a covering $I = \bigcup_{\alpha \in A} I_\alpha$ is admissible if $A$ is finite.

~~In this case the category of sheaves agrees (I think) with the category of sheaves on the ordinary topological space $\mathbb{R}$ with its Euclidean topology. If necessary, I can provide a proof.~~

For another example (which is actually quite similar), let $\mathrm{Sp}(A)$ be the rigid analytic space associated to a $K$-affinoid algebra $A$ where $K$ is a non-archimedean field. It is a G-topological space whose points are the maximal ideals of $A$, whose admissible open subsets are the affinoid domains and whose admissible coverings are the finite ones.
~~Then the sheaves on $\mathrm{Sp}(A)$ agree with sheaves on $\mathscr{M}(A)$, the Berkovich space associated to $A$ with its Berkovich topology. There is a canonical map $\mathrm{Sp}(A) \to \mathscr{M}(A)$ identifying the affinoid domains of $\mathrm{Sp}(A)$ with the affinoid domains of $\mathscr{M}(A)$ in the Berkovich sense.~~
As remarked by Piotr Achinger below, the Berkovich spectrum should be replaced with the Huber spectrum here.

[BGR84] *S. Bosch*, *U. Güntzer* and *Reinhold Remmert*, Non-archimedean analysis. Berlin etc.: Springer-Verlag (1984; Zbl 0539.14017)

[MLM92] *S. Mac Lane* and *I. Moerdijk*, Sheaves in geometry and logic: a first introduction to topos theory. New York etc.: Springer-Verlag (1992; Zbl 0822.18001)