I'm reading DeWitt's Supermanifolds and in part I he addresses the theme of the "various" existing superspaces. Concretely in p.10 he mentions the following three:
$ \mathbb{R}^{n|\nu} $ spanned by $ (x^1,...,x^n,\xi^1,...,\xi^{\nu}) $ with $ \xi $ being the generators of the Grassmann algebra.
$ \Lambda^{n+\nu} $ the Grassmann algebra equipped with coordinates $ (z^1, ..., z^{n+\nu}) $ and
$ \mathbb{R}^n_c \times \mathbb{R}^{\nu}_a $ the space of even and odd vectors in Euclidean superspace equipped with a basis $ (u^1,...,u^n,v^1,...,v^{\nu}) $ where the $u$'s are even and the $v$'s are odd.
Later on (p.12), he states that the target space of the coordinates on a supermanifold is (3). My question is why don't we use (1) instead? Wouldn't it be better since we can reduce it to Euclidean $ \mathbb{R}^n $ in the limit $ \xi =0 $? It also has many other properties of the Euclidean non-supersymmetric space unaltered. Why use (3) instead?
Also it seems that he ends up using (1) later since in part II where defining integration on mixed spaces he introduces coordinates on a supermanifold as $ x^A = (x^a, \xi^{\alpha}) $ which is only possible if the target space was (1). So what is the actual target space of supermanifolds? Is it (1) or is it (3)?
