# What changes in the sheaf theory of topological spaces with the “étale topology”?

The customary site structure on the category of topological spaces has covering families given by open covers. What "happens" if we refine this topology and let any jointly surjective family of local homeomorphisms be covering?

Refining the Zariski topology has lots of interesting consequences, so I wonder what kind of thing happens for topological spaces.

Absolutely nothing changes. The category of etale sheaves on a topological space $$X$$ is equivalent to the category of ordinary sheaves on $$X$$. The reason is that ordinary open subsets (i.e., maps $$U\to X$$ which are homeomorphisms to some open subset of $$X$$) are cofinal in all etale maps to $$X$$, since you given any etale map $$p:Y\to X$$ you can cover $$Y$$ by open subset on which $$p$$ is a homeomorphism to its image. So, any etale sheaf is determined by the values it takes just on open subsets of $$X$$, and those values just need to form an ordinary sheaf.