Lemma 65.25.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then $f(X_{\text{ft-pts}}) \subset Y_{\text{ft-pts}}$.

**Proof.**
Take $x \in X_{\text{ft-pts}}$. Represent $x$ by a locally finite type morphism $x : \mathop{\mathrm{Spec}}(k) \to X$. Then $f \circ x$ is locally of finite type by Lemma 65.23.2. Hence $f(x) \in Y_{\text{ft-pts}}$.
$\square$

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