Suppose given a topological space $(X,\mathcal O_X)$. Then we know its soberification is defined as a tuple $(X^s,\mathcal O_{X^s})$, where
- The set $X^s$ is the set of all irreducible closed subset of $X$.
- For an open set $U$, define $U^s=\{A \in X^s \mid A \cap U \neq \emptyset\}$. Then the set of all open sets on $X^s$ is defined as: $$\mathcal O_{X^s}= \{U^s \mid U\in \mathcal O_X\}.$$
Among other things, it is well-known that the two set of open sets are isomorphic, i.e., $\mathcal O_X \cong \mathcal O_{X^s}$. Here, by isomorphism, I mean there is a bijection between the sets of open sets.
Now my question is: do we have any relationship with the Borel sigma-algebras generated by the two sets of open sets? For instance, is it true that $\sigma(\mathcal O_X) \cong \sigma(\mathcal O_{X^s})$? Here, $\sigma(\mathcal O_X)$ is the smallest sigma-algebra generated by the collection $\mathcal O_X$.
Thank you.
PS: The above construction is detailed in the document http://stacks.math.columbia.edu/download/topology.pdf; see Lemma 8.14 on Page 10.