I am reading Clausen–Scholze’s Condensed Mathematics notes. In the proof of Proposition 9.6, the following statement appears to be used:
Let $f: Y\to Z$ be a morphism of schemes which is locally on the source an open immersion. Then $f$ is an open immersion (resp. isomorphism) iff $f$ is injective (resp. bijective) on the underlying point sets.
Is this general statement true, or is my understanding of the proof somehow wrong? (In the notes, $Y=X^{\mathrm{ad}}$ and $Z=X^{\mathrm{ad}/R}$.)
Yes, this is true. One characterization of an open immersion of schemes is an open immersion of topological spaces $f:Y\to Z$ such that the induced map on stalks $\mathcal{O}_{Z,f(y)}\to \mathcal{O}_{Y,y}$ is an isomorphism for all $y\in Y$. The local condition specified by your authors gives the condition on sheaves, and also that $f$ is open. As an injective open map is a homeomorphism on to its image and thus a topological open immersion, the injectivity condition specified by your authors is just what you need to make your conclusion that $f$ is an open immersion.