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Soft question: Union of infinitely many closed sets
BTW: In your example with growing disks you state that the union of growing disc cover the whole plane which is open. But it is also closed (clopen), just like the empty set.
Can a Rubik's cube be mapped knowing only two sides?
Well, strictly speaking it shows that knowing two opposite sides does not suffice. "Looking at only two sides" seems to imply that the sides are choosen by the solver, so he might choose adjacent sides (Ok, we now know that this case is also closed).