An *immersion* (or *locally closed immersion*) of schemes is a morphism that can be factored as , where is a closed immersion and is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

**Example.** Let be a nonreduced scheme, and let 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.

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