Supermanifold

1 Classical manifold

Ordinary manifold of dimension is a topological space with a sheaf of commutative algebras called , satisfying the following property. Every point has a neighborhood such that is the algebra of functions on some open subset .

This can be also said in another way:

A manifold structure on a topological space is a rule which to every small enough open set associates:
an equivalence class of pairs where is an open subset of and is a homeomorphism
with the following equivalence relations: if exists a diffeomorphism such that
Moreover, it is required that, every time when are two open subsets of , and is in the class associated to , then is in the class associated to .

2 Supermanifold

A supermanifold structure on a topological space is a rule which to every small enough open set associates:
an equivalence class of pairs where is a superdomain of dimension and is a homeomorphism from the body of to
with the following equivalence relations: iff exists an isomorphism of superdomains such that
Moreover, it is required that, every time when are two open subsets of , and is in the class associated to , then belongs to the class associated to .