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I have the following Grothendieck pretopologies on the category of schemes.

The first one, the covers are families of morphisms $\{ U_i \to X \}$ such that for every point $x \in X$ there exists some $i$ and some $u \in U_i$ over $x$ such that $[k(u):k(x)] = 1$ (notice I haven't asked anything else of the morphisms $U_i \to X$; they don't have to be ├ętale, or even of finite type).

The second one, the covers are finite families of morphisms $\{ V_i \to X\}$ of finite type such that $\amalg V_i \to X$ is surjective and each $V_i \to X$ is an immersion (i.e. a composition of closed and open immersions).

If the schemes are noetherian they should give rise to the same category of sheaves.

Obvious choices in my mind for naming the first one is "completely decomposed" (for obvious reasons) or "discrete" (because every cover is refinable by the cover $\{ x \to X \}_{x \in X}$), but both of these exist already. The second I would call the "open-closed topology" but this exists as well.

Any suggestions? Does there already exists a name in the literature?

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There is already a "completely decomposed" topology floating out there: en.wikipedia.org/wiki/Nisnevich_topology –  stankewicz Apr 6 '12 at 12:38
    
I guess I should make a note about Nisnevich's topology vs. your pretopology (and I'm not any sort of expert in Grothendieck topologies so caveat lector). Nisnevich explicitly requires his morphisms be etale so that his topology a) can be used for studying class groups of tori (or so I was told by Cristian Gonzalez-Aviles) and b) lies strictly between the etale topology and the zariski topology. Do you have any particular reason for not placing restrictions on your morphisms? –  stankewicz Apr 6 '12 at 12:44
    
The motivation for my definition is the following: I have a particular functor $F$ in mind. I can show that for every scheme $X$ in the category of schemes that I am working, the canonical morphism $F(X) \to \prod_{x \in X} F(x)$ is injective. The group on the right is precisely the sheafification of $F$ for the topology described in the question, and this language makes the statements and proofs much cleaner. For example stating this morphism is a monomorphism is just saying that $F$ is separated for the topology described in the question. –  name Apr 7 '12 at 8:56

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