quetion analoguing to this mainly talk about problem about bijective.
Exercise 3.10 in Hartshorne.Let $f: X\rightarrow Y$ be a morphism of schemes and let $y\in Y$, then $X_y=X\times_Y \operatorname{Spec}k(y)$ is homeomorphic to $f^{-1}(y)$ with the induced topology.
my question is: how to prove $X_y\rightarrow f^{-1}(y)$ sents open set to open set?i.e. the inverse is continuous.
In fact,I have no idea.I think it is just bijective.there is no infromation about the inverse.
many thanks in advance.