then we do some mumble mumble point set topology ergodic theory thing
hrmm. construct for any ordinal α S_α
S_0=S
S_{α+1} is S_α minus all its isolated points
S_λ is the intersection of S_α α<λ
S_α is always closed, nonempty, and infinite
Follow
S_α is also eventually constant if for no other reason than that eventually you run out of points. call the resulting set T
pretty sure because T is perfect and this is a nice enough topological space T has cardinality continuum
and one point in this space always gives you an infinite number because aperiodic
(note this reasoning does not hold if there exists a periodic tiling, and I feel like it's not too difficult to construct counterexamples here)
i think this reasoning still works if we're talking tiles where there's only a finite number of ways to glue 2 tiles together sensibly
you don't quite get the group action but you do still get a compact topology on all the ways to build a tesselation around some base tile