Are representables on the étale site on topological space sheaves? The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme.
Are representables on the étale site on topological space sheaves? Why should one expect this to be the case for schemes?
 A: Yes, this is true; the point is that étale maps locally have continuous inverses.  An étale cover of a topological space $X$ is a space $Y$ together with a map $p:Y\to X$ which is surjective and a local homeomorphism.  To say representables are sheaves is to say that if $Z$ is a topological space and $f:Y\to Z$ is a continuous map such that $p(x)=p(y)$ implies $f(x)=f(y)$, then there is a unique continuous map $g:X\to Z$ such that $gp=f$.  Clearly there is a unique such map of sets $g$, so it suffices to check that this $g$ is continuous.
Fix a point $x\in X$.  Since $p$ is surjective, there is some $y\in Y$ such that $p(y)=x$.  Since $p$ is a local homeomorphism, there is an open neighborhood $U\subseteq Y$ of $y$ such that $p(U)$ is open in $X$ and $p$ restricts to a homeomorphism $p_U:U\to p(U)$.  For all $z\in p(U)$, we then have $g(z)=g(p(p_U^{-1}(z)))=f(p_U^{-1}(z))$.  Thus $g|_{p(U)}=f\circ p_U^{-1}$ and is hence continuous.  Thus $g$ is continuous in a neighborhood of any point of $X$, and hence continuous on $X$.
