Well, I can't believe nobody did anything with what I gave you. You make me sad. So I take answer away. you should have tried the solution to the universe was right there in front of your eyes. But it always has been and you never saw it then either. You will see I'm not kidding what I handed you was the absolute mathematical proof that it is all one thing we all come from central black hole all the galaxies are stacked like pancakes on top of one another. I wasn't trying to be a smart ass I really thought you would enjoy solving the universe. But you all have been filled with so much bad science that you did not believe you could do it even when the answer was laid in front of you. I was even going to let the person who did it publish it under their own name. You could have done it any one of you it is so simple. When you see it you will kick yourself. If you get anything from this understand that you can understand the universe. It simple and beautiful a child could have seen what I gave you. But you cant see it. Ask your self why you could not see what I will publish tomorrow when it was right in front of you. Then ask what else you cant see. I need you to be able to see and learn the real universe this is the first step. Im not giving up must be away to clear all this crap out of your head so you can think again.
Ok so you obviously have an open set UU of the refinement that contains the origin and which is contained in the original origin-centered ball BB or radius 1/31/3. You're worried that UU has to intersect infinitely many refinement sets each of which is confined to its own spine, but thats not necessarily the case. A refinement set, WW, that intersects UU could be contained in BB, and so WW could be the disjoint union of pieces of the interiors of all the spines (subject to the star-finiteness condition). Thus WW could "bridge the gap" between UU and finitely many of the refinements of the 3/43/4 balls. After that you do the same thing again: take a subset, VV, of BB that bridges the gap between WW and a few more (finitely many) refinements of the 3/43/4 balls, but also such that VV is disjoint from UU etc.
P.S. On a manifold, you can always take a refinement with an injective pairing, taking an element of the refinement to a superset element of the original cover. This would prohibit the above idea since WW would have to find its own superset element of the original cover (BB already being taken by UU). I don't know if there's a name for this property and I don't remember what manifold axioms are used to get it -- it would be interesting and reassuring to see where it fails in the case of the hedgehog space.