I suppose that this is false in general: A map of topological spaces $f:X\to Y$ is closed iff it is locally closed, i.e. there is an open covering $\{V_i\hookrightarrow X\}$ such that each $f|_{V_i}$ is closed.
Let $f:X\to Y$ be a map of noetherian schemes, so in particular a map of noetherian topological spaces equipped with the Zariski topology. Does "locally closed" imply "closed" in this case?
(Actually, I want to test if $f$ is a closed immersion and I know that it is "locally" (in the above sense) a closed immersion.)