  Inverse limits and profinite groups are used in a quantum mechanical context.
Two cases are considered. A quantum system with positions in the profinite
group ${\mathbb Z}_p$ and momenta in the group ${\mathbb Q}_p/{\mathbb Z}_p$;
and a quantum system with positions in the profinite group ${\hat {\mathbb Z}}$
and momenta in the group ${\mathbb Q}/{\mathbb Z}$. The corresponding
Schwatz-Bruhat spaces of wavefunctions and the Heisenberg-Weyl groups are
discussed. The sets of subsystems of these systems are studied from the point
of view of partial order theory. It is shown that they are directed-complete
partial orders. It is also shown that they are topological spaces with $T_0$
topologies, and this is used to define continuity of various physical
quantities. The physical meaning of profinite groups, non-Archimedean metrics,
partial orders and $T_0$ topologies, in a quantum mechanical context, is
discussed.
