Lemma 39.19.3. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s and t quasi-compact and flat and U quasi-separated. Let W \subset U be quasi-compact open. Then t(s^{-1}(W)) is an intersection of a nonempty family of quasi-compact open subsets of U.
Proof. Note that s^{-1}(W) is quasi-compact open in R. As a continuous map t maps the quasi-compact subset s^{-1}(W) to a quasi-compact subset t(s^{-1}(W)). As t is flat and s^{-1}(W) is closed under generalization, so is t(s^{-1}(W)), see (Morphisms, Lemma 29.25.9 and Topology, Lemma 5.19.6). Pick a quasi-compact open W' \subset U containing t(s^{-1}(W)). By Properties, Lemma 28.2.4 we see that W' is a spectral space (here we use that U is quasi-separated). Then the lemma follows from Topology, Lemma 5.24.7 applied to t(s^{-1}(W)) \subset W'. \square
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