  Motivated by analogies with basic density theorems in analytic number theory,
we introduce a notion (and variations) of the homological density of one space
in another. We use Weil's number field/ function field analogy to predict
coincidences for limiting homological densities of various sequences
$\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ of spaces of $0$-cycles on manifolds $X$.
The main theorem in this paper is that these topological predictions, which
seem strange from a purely topological viewpoint, are indeed true.
  The obstacle to proving such a theorem with current technology is how to deal
with the combinatorial complexity of all possible "collisions" of points, this
problem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ of
configuration spaces. To overcome this obstacle we develop a method that uses
the Bj\"orner--Wachs theory of lexicographic shellability from algebraic
combinatorics to study such problems. As a consequence we derive new
homological stability theorems for broad classes of $0$-cycles on manifolds.
Even in the classical case $(m,n)=(1,2)$ this gives a new, simplified proof of
classical results, and also of recent theorems of Church and others.
