Lemma 01RE. Let $j:U→X$ be an open immersion of schemes. Then U is scheme theoretically dense in X if and only if $\mathscr O_X→j_*\mathscr O_U$ is injective
I am surprised by this lemma on Stacks Project as by definition, an open immersion is an homeomorphism $U \simeq V \subset X$ where $V$ is open topological space, and $j_*\mathscr O_U \cong \mathscr O_X$
So as a consequence, every image of open immersion over $X$ is scheme theoretically dense in $X$.
Does it make sense?
Thank you.