Étale space of a presheaf ($\mathrm{Et}(P)={\coprod }_{x\in X}{P}_x$)

The space of germs of a presheaf has basic opens given by the germs of one section over an open set. Projection to its base point is a local homeomorphism. Continuous sections of this projection give sheafification. For the constant presheaf of sets, the basic opens are $U\times \{t\}$, giving the ordinary product with a discrete fibre.