H topology

In algebraic geometry, the "h" topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the "qfh" and "cdh" topologies.
Definition.
Define a morphism of schemes to be "submersive" or a "topological epimorphism" if it is surjective on points and its codomain has the quotient topology, i.e., a subset of the codomain is open if and only if its preimage is open. A morphism is "universally submersive" or a "universal topological epimorphism" if it remains a topological epimorphism after any base change.
Voevodsky defines the "h" topology on the category of schemes to be the topology associated to finite families formula_1 of morphisms of finite type such that formula_2 is a universal topological epimorphism.
The "qfh" topology is associated to families as above, with the further restriction that each formula_3 must be quasi-finite.
"cdh" topology.
While defined on all schemes, the "h" and "qfh" topology are only ever used on Noetherian schemes. The "h" topology has various non-equivalent extensions to non-Noetherian schemes including the "ph" topology and the "v" topology.
The proper "cdh" topology is defined as follows. Let be a proper morphism. Suppose that there exists a closed immersion . If the morphism is an isomorphism, then "p" is a covering morphism for the "cdh" topology. The "cd" stands for "completely decomposed" (in the same sense it is used for the Nisnevich topology). An equivalent definition of a covering morphism is that it is a proper morphism "p" such that for any point "x" of the codomain, the fiber "p"−1("x") contains a point rational over the residue field of "x".
The "cdh" topology is the smallest Grothendieck topology whose covering morphisms include those of the proper "cdh" topology and those of the Nisnevich topology.
Properties.
The "h" topology combines a number of useful properties of its various "sub"topologies. Since if is finer than the Zariski topology, "h"-locally every scheme is affine. Since it is finer than the Nisnevich topology, "h"-locally regular immersions look like zero sections of vector bundles. It is also finer than the étale topology and the fppf topology.
In a different direction, it is finer than the "qfh" topology, so "h" locally, algebraic correspondences are finite sums of morphisms. Finally, every proper surjective morphism is an "h" covering, so in any situation where de Jong's theorem on alterations is valid, "h" locally all schemes are regular.
Relation to v-topology.
The v-topology (or universally subtrusive topology) is equivalent to the h-topology on Noetherian schemes. On more general schemes, the v-topology has more covers.