Lemma 109.24.1. There exists a morphism $f : X \to Y$ of finite presentation between affine schemes and a locally closed subset $T$ of $X$ such that $f(T)$ is not a finite union of locally closed subsets of $Y$.

Proof. See discussion above. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).