Noah Schweber said [here](https://math.stackexchange.com/questions/1902347/grothendieck-topology-and-relation-with-usual-topologies) the following:



> 
> Why would you want a notion of sheaf theory for objects more general
>  than topological spaces? Well, the original motivation (to my
>  understanding) was to develop a notion of etale cohomology for
>  schemes; so if you care about schemes, you should care about sites.
> 
> 
> 


Question : In what sense does Grothendieck topologies are in relation with Etale cohomology of Schemes? 


Any other explanation for "why would you want a notion of sheaf theory for objects more general than topological spaces?" is also welcome. 



Why care about Grothendieck topology?
Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only.


In the category of topological manifolds, an etale cover of $X$ is a surjective local homeomorphism $Y\rightarrow X$. In the etale topology on varieties, this is "morally true" as well with local homeomorphism, replaced by a surjective morphism, defining isomorphism of tangent cones. In the schemes the cones are further replaced by henselisations of local rings...


Note that there are numerous further examples of useful Grothenidieck topologies that are not topologies...


