In differential geometry, an etale bundle on a manifold $X$ is a bundle whose projection is a local homeomorphism. This can be equivalently presented as a sheaf of rings on $X$.
This can be generalised to topological spaces: There is an adjunction between $Top/X \dashv PShf(X)$ where the left adjoint is the space of sections and the right adjoint is the space of germs. This restricts to an equivalence between $Et(X) \dashv Shf(X)$.
The definition I've seen of schemes starts from a sheaf that is valued in $Ring$ rather than $Set$: taking $X$ to be any topological space, it is a sheaf of rings on $X$ that is locally affine - i.e. isomorphic to the spectrum of some commutative ring.
Is there equivalent a characterisation of this as an etale space? ie for every scheme $S$ is there an etale space $S'$ such that its space of sections is isomorphic to the scheme? If this isn't possible, is there a characterisation of schemes for which this is possible?