Lemma 100.9.6. If $f : \mathcal{X} \to \mathcal{Y}$ is an immersion, then $|f| : |\mathcal{X}| \to |\mathcal{Y}|$ is a homeomorphism onto a locally closed subset. If $f$ is a closed, resp. open immersion, then $|f|$ is closed, resp. open.

**Proof.**
Omitted.
$\square$

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