# Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic geometry, except sometimes as a word for an immersion of varieties. And the notion of an "immersion" of schemes, especially an "open immersion," seems much more similar to the topologists' "embedding" than their "immersion." [Closed immersions at least have the somewhat flimsy rationale that the scheme structure does not depend solely on the choice of subset.]

Does anyone know of a good reason, other than cultural momentum, to use the word "immersion" rather than "embedding"?

[Note: this has come up in Ravi Vakil's blog on his Algebraic Geometry notes.]

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What do you call a morphism $f:X \rightarrow Y$ such that $f$ carries $X$ homeomorphically onto an open subset $U$ but the induced map $X \rightarrow U$ is not necessarily a scheme isom? I like to call it an open embedding. That is, "embedding" encodes the topological aspect, and "immersion" means one keeps appropriate track of the structure sheaves too. The distinction is invisible in differential geometry since an immersion between connected manifolds has the manifold structure on the source uniquely determined by the topology of the situation and the manifold structure on the target. –  BCnrd Dec 7 '10 at 4:08
It's a little confusing that things are the other way round in differential geometry -- an immersion need not be an embedding. (That's a distinction that's missing in algebraic geometry, at least for "closed immersions"!) –  Dave Anderson Dec 7 '10 at 7:21
something I find rather irritating is the fact that the thing I would think ought to be called an ''open immersion'' is called instead an ''etale morphism''... –  Vivek Shende Dec 7 '10 at 15:11
Vivek--In that case, you should also be irritated that we have 'covering space' in our vocabulary. –  Keerthi Madapusi Pera Dec 7 '10 at 16:30
Dear Dave: In EGA IV$_4$, sections 16.9 and 19, such maps are called "regular immersions"! Dear Vivek: etale maps that are injective on geometric points are open immersions, so Keerthi's comment seems quite apt and you may be amused to know that Grothendieck's original attempt at defining the etale topology was via finite etale covers of Zariski opens (before Artin convinced him to switch to etale surjections). –  BCnrd Dec 7 '10 at 17:56