Lemma 67.34.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $n \geq 0$. Assume $f$ is locally of finite type. The set
is open in $|X|$.
Proof. Choose a diagram
where $U$ and $V$ are schemes and the vertical arrows are surjective and étale, see Spaces, Lemma 65.11.6. By Morphisms, Lemma 29.28.4 the set $U_ n$ of points where $h$ has relative dimension $\leq n$ is open in $U$. By our definition of relative dimension for morphisms of algebraic spaces at points we see that $U_ n = a^{-1}(W_ n)$. The lemma follows by definition of the topology on $|X|$. $\square$
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