  In this article, we present a novel theory of locally semialgebraic
superspaces along with Nash supermanifolds. By adapting Batchelor's theorem to
our framework, we show that all locally semialgebraic superspaces and affine
Nash supermanifolds can be derived from appropriate vector bundles. Our
analysis of Nash supermanifolds involves an investigation of a topology, where
closed sets are defined by Nash subsets. Within this context, we establish the
softness of the structure sheaf for an affine Nash manifold. Moreover, we
establish that under this topology, the higher cohomology of quasi-coherent
sheaves on an affine Nash manifold vanishes completely. These results open up
new directions for cohomological studies on Nash manifolds.
