There is a very basic theorem for the Zariski topology.
Let X = Spec(R) and Y=Spec(R/I) for I some reduced ideal. Y obtains a topology two ways, one is the subspace topology as a subset of X and another as the spectrum of a ring. These topologies are the same by the correspondence between ideals of R containing I and ideals in R/I.
Is there a close statement to this in the etale toplogy? There are two natural ways to understand open sets on Y, those which come from etale neighborhoods of X base changed to Y and those which are etale neighborhoods of Y.
I did a computation today in a very special case and it seems that both of these topologies seem to be 'the same'.
Does anyone know if this statement is true in a general context and where I might locate this resource?