ScalingStacks

Proposition 2.94. Let $\phi:X\rightarrow Y$ be a morphism of schemes of locally finite type over $\spec k$. Then it induces a continuous map $\phi^{\mathrm{an}}:X^{\mathrm{an}}\rightarrow Y^{\mathrm{an}}$. And $\phi$ is (1) separated, (2) injective, (3) surjective, (4) an open immersion and (5) an isomorphism if and only if $\phi^{\mathrm{an}}$ has the same property. ([Ber, Proposition 3.4.6])