A $K$-analytic space $X$ is called Stein if the natural map

$$X\to Spec^{an}(\Gamma(X,{\cal O}_{X}))$$

is a homeomorphism. Here $\Gamma(X,{\cal O}_{X})$ is considered as a topological $K$-algebra. This definition is equivalent to the standard one. Let us call the projection $\pi:X\to B$ Stein if for any $b\in B$ there exists a fundamental systems of neighborhoods $U_{i}$ of $x$ such that $\pi^{-1}(U_{i})\subset X$ is a Stein domain. If $\pi$ is Stein then we can reconstruct $(X,{\cal O}_{X})$ and $\pi$ from the space $B$ endowed with the sheaf $\pi_{*}({\cal O}_{X})$ of topological $K$-algebras.

Proof.³³ 3 We thank to Ofer Gabber for suggesting the proof below It suffices to prove that $n$-dimensional cells are dense in $B$, where $n=\dim X$. For any open $U\subset B,\,\,U\neq\emptyset$ we have $H_{c}^{n}(U,\pi_{\ast}(\Omega^{n}_{X}))\simeq H_{c}^{n}(\pi^{-1}(U),\Omega^{n}_{X})$.

The last group is nontrivial, because for any non-empty open $V\subset X^{an}$ the integration map $\int:H_{c}^{n}(V,\Omega^{n}_{X})\to K$ is onto. Therefore $\dim(U)\geq n$. $\blacksquare$

All the examples in Sections 4.2.1–4.2.5 (except Section 4.2.3) have Stein property.