Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober? 



Morphism of schemes with non-sober image
Let $X$ be a disjoint union of $\operatorname{Spec}\mathbb F\_p$ over primes $p$ and consider the obvious map $X\to\operatorname{Spec}\mathbb Z$. The image $\operatorname{Spec}\mathbb Z\setminus\{(0)\}$ is not sober.


