Not every open immersion is an open immersion

An immersion (or locally closed immersion) of schemes is a morphism f \colon X \to Y that can be factored as X \to U \to Y, where X \to U is a closed immersion and U \to Y is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

Example. Let X be a nonreduced scheme, and let X_{\text{red}} \to X be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism. \qedsymbol

Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.

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