better suited to study with the tools of standard differential geometry and locally ringed spaces. A supermanifold M of dimension (p,q) is a topological space M with a sheaf of superalgebras, usually denoted OM or C∞(M), that is locally isomorphic to , where the latter is a Grassmann (Exterior) algebra on q generators. A supermanifold M of dimension (1,1) is sometimes called a super-Riemann surface. Historically, this approach is associated with Felix Berezin, Dimitry Leites, and Bertram Kostant. Concrete: as a smooth manifold A different definition describes a supermanifold in a fashion that is similar to that of a smooth manifold, except that the model space has been replaced by the model superspace . To correctly define this, it is necessary to explain what and are. These are given as the even and odd real subspaces of the one-dimensional space of Grassmann numbers, which, by convention, are generated by a countably infinite number of anti-commuting variables: i.e. the one-dimensional space is given by where V is infinite-dimensional. An element z is termed real if ; real elements consisting of only an even number of Grassmann generators form the space of c-numbers, while real elements consisting of only an odd number of Grassmann generators form the space of a-numbers. Note that c-numbers commute, while a-numbers anti-commute. The spaces and are then defined as the p-fold and q-fold Cartesian products of and . Just as in the case of an ordinary manifold, the supermanifold is then defined as a collection of charts glued together with differentiable transition functions. This definition in terms of charts requires that the transition functions have a smooth structure and a non-vanishing Jacobian. This can only be accomplished if the individual charts use a topology that is considerably coarser than the vector-space topology on the Grassmann algebra. This topology is obtained by projecting down to and then using the natural topology on that. The resulting topology is not Hausdorff, but may be termed "projectively Hausdorff". That this definition is equivalent to the first one is not at all obvious; however, it is the use of the coarse topology that makes it so, by rendering most of the "points" identical. That is, with the coarse topology is essentially isomorphic to Properties Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a