Its objects can be informally thought of as étale open subsets of X.
An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets.
A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces.
Suppose that U → X is an object of Ét(X) and that Ui → U is a jointly surjective family of étale morphisms over X.
Every torsion sheaf is a filtered inductive limit of constructible sheaves.
In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms.
An étale cover of an affine scheme X can be defined as a jointly surjective family {uα : Xα → X} such that the set of all α is finite, each Xα is affine, and each uα is étale.
In the étale topology, there are strictly more open neighborhoods of x, so the correct analog of the local ring at x is formed by taking the limit over a strictly larger family.