Since the universe is expanding, does anything ever occupy the same point in space? Let's say we can observe expansion in a supercluster.
We define origin of our frame of reference at the center of the supercluster.
We observe an object/atom at point A at time T. The object is motionless relative to the origin.
We wait for expansion until T+ΔT and again observe the object.
Is the object at A or somewhere else?
 A: In the classical description of general relativity, the spacetime points constitute a smooth manifold with local coordinates $x^{\mu}$.  In order to compute distances and intervals between points, an extra piece of information is needed, namely the metric tensor field $g_{\mu\nu}$(x).  Space expansion can be thought of as not moving the points around, but rather as simply a change of the spatial part of the metric tensor with time.
For example, if we write the spacetime coordinates as $x^{\mu}$ = (t, $x^i$) where i=1,2,3 then we can write the metric of a spatially expanding universe as  $$ds^2 = c^2dt^2-a(t)^2g_{ij}dx^idx^j$$ where the spatial metric components $g_{ij}$ do not depend on time.
So spatial expansion is a change in the metric tensor field, rather than any "motion" of the spacetime points themselves.
An alternative way to think of this is that points of a spacetime manifold in isolation (i.e. with no other physical fields defined) do not have any physical  significance.  This is connected with Einstein's hole argument. 
A: This seems to be dependent on the manner in which the super-cluster expands.
If expansion, mass, and energies expand perfectly synchronous relative to each other and the point defined as "center", then the "center" remains relative to all other points. We know this not to be the case, therefore, the "center"will constantly relocate in relation to all other points of the super-cluster as defined by the mechanisms affecting all other points.
Where's the center of the vortex in the toilet bowl? Constantly in flux due to motions of mass relative to it.
