  In this work we introduce a new combinatorial notion of boundary $\Re C$ of
an $\omega$-dimensional cubing $C$. $\Re C$ is defined to be the set of
almost-equality classes of ultrafilters on the standard system of halfspaces of
$C$, endowed with an order relation reflecting the interaction between the
Tychonoff closures of the classes.
  When $C$ arises as the dual of a cubulation -- or discrete system of
halfspaces -- $\HH$ of a CAT(0) space $X$ (for example, the Niblo-Reeves
cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we
show how $\HH$ induces a function $\rho:\bd X\to\Re C$. We develop a notion of
uniformness for $\HH$, generalizing the parallel walls property enjoyed by
Coxeter groups, and show that, if the pair $(X,\HH)$ admits a geometric action
by a group $G$, then the fibers of $\rho$ form a stratification of $\bd X$
graded by the order structure of $\Re C$. We also show how this structure
computes the components of the Tits boundary of $X$.
  Finally, using our result from another paper, that the uniformness of a
cubulation as above implies the local finiteness of $C$, we give a condition
for the co-compactness of the action of $G$ on $C$ in terms of $\rho$,
generalizing a result of Williams, previously known only for Coxeter groups.
