  We discuss the problem of finding an analogue of the concept of a topological
space in supergeometry, motivated by a search for a procedure to compactify a
supermanifold along odd coordinates. In particular, we examine the topologies
naturally arising on the sets of points of locally ringed superspaces, and show
that in the presence of a nontrivial odd sector such topologies are never
compact. The main outcome of our discussion is that not only the usual
framework of supergeometry (the theory of locally ringed spaces), but the more
general approach of the functor of points, need to be further enlarged.
