In Daping Weng's A Categorical Introduction to Sheaves, in the paragraph preceding section 3.2 (page 8), the author says
A sheaf space [= Étalé space] almost looks like a sheaf...
I would very much like to understand why the author says this and how I can see it for myself. In what sense does the Étalé space "almost look like a sheaf"?
The only bit I can see for myself goes through the agricultural analogy that the Étalé space gathers germs into stalks, which takes us a step closer to sheaves. Indeed, the author says the only remaining thing to do is to "tie" the stalks together, which is supposedly what the sheaf-of-cross-sections functor does. However, I how does "each cross section over an open set is open and homeomorphic to the underlying set" make the Étalé space "look like a sheaf"?