Set-valued homology

We establish homology groups based on a class of set-valued mappings and prove they are isomorphic to the classical homology groups. Any set-valued mapping in this class induces homomorphisms between homology groups of two topological spaces. Set-valued homology can be used to study set-valued mappings between two spaces whose homology groups are already known. For instance, it follows a version of the Kakutani Fixed Point Theorem in the nonconvex and nonacyclic case. We can get some other properties of set-valued mappings. Besides, applying set-valued homology, we can get a result for ordinary single-valued mappings, which is analogous to the Vietoris Mapping Theorem.