Smooth topology
In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf.
To understand the problem that motivates the notion, consider the classifying stack over. Then in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of to be more like that of as the ring should classify line bundles. Thus, the cohomology of should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.
