What is an étale covering I'm reading Buouville's book: Complex Algebraic Surfaces and in particular i'm interested about the part exposing nice properties of surfaces of general type.
During the lecture i'have found the term étale covering of a surface but on Bouville there's no references about that.
On the web there are lot of different defintions and i need something more precise. Any suggestion about references or some hint about this? 
 A: My understanding of étale topologies is from a viewpoint of categories, but I believe it will be the same situation for surfaces.
For a topology we need open sets, which is usually a nice natural object.  But if you consider something such as a category, how could you go about defining open sets?
Grothendieck answered this for us, he introduced a whole class of topologies (Grothendieck topologies) and étale topology is one of these.  A Grothendieck topology is given by a function t which assigns to each object U of a category C a collection t(U) consisting of families of morphisms {p_i:U_i --> U} for i in some index set such that it satisfies some properties. Then the families in t(U) are called covering families for U.  
So we can see it as letting the class of morphisms play the role of open sets in analogy to topological spaces. 
Now, we have a generic Grothendieck topology, for the étale one we consider étale maps, ie flat and unrammified maps.  Therefore for the families {p_i:U_i --> U} we require p_i to be étale maps!
