Just curious if anyone has ever studied what happens when a topology gains new members in the underlying set. How is it incorporated into the existing subsets whose union and intersection are included in the topology? It seems to me that assuming the universe expanded from a singularity, then more space with more time would add more elements to the underlying set, which is the universe as a whole. When we engrave a coordinate system on this topology (as with manifolds), Do the dimensions grow to incorporate the new elements of (spacetime?)? Are new subsets born which must be included? What? I'm not sure this question belongs here, but it seems it should be a consideration about the basic elements of an expanding spacetime. Shouldn't this be a consideration of quantum gravity? Any help is appreciated.