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 étale, 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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