  Some models of set theory are given which contain sets that have some of the
important characteristics of being geometric, or spatial, yet do not have any
points, in various ways. What's geometrical is that there are functions to
these spaces defined on the ambient spaces which act much like distance
functions, and they carry normable Riesz spaces which act like the Riesz spaces
of real-valued functions. The first example has a family of sets, each one of
which cannot be empty, but not in a uniform manner, so that it is false that
all of them are inhabited. In the second, we define one fixed set which does
not have any points, while retaining all of these geometrical properties.
