The Stacks project

Lemma 65.12.4. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. Let $F_1$, $F_2$ be locally closed subspaces of $F$. If $F_1 \subset F_2$ as subfunctors of $F$, then $F_1$ is a locally closed subspace of $F_2$. Similarly for closed and open subspaces.

Proof. Let $T \to F_2$ be a morphism with $T$ a scheme. Since $F_2 \to F$ is a monomorphism, we see that $T \times _{F_2} F_1 = T \times _ F F_1$. The lemma follows formally from this. $\square$

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