# Is finite union of locally closed subscheme, a scheme

Let $X$ be a projective, noetherian $k$-scheme for an algebraically closed field $k$ of characteristic zero. Let $Y_1,...,Y_r$ be locally closed subschemes (open subschemes of closed subschemes) of $X$. Does the scheme structure on $X$ necessarily induce a scheme structure on $Y_1 \cup ... \cup Y_r$?

• One motivation for introducing this notion is Chevalley's theorem, that the image of a morphism is a constructible set (example: consider $(x,y)\mapsto (x,xy)$). Sep 20, 2017 at 14:09