jlg@lanl.ARPA (01/12/85)
> I would like to hear a discussion of the implications of a closed and/or > bounded (not the same thing) universe on such issues. I think there is > something very interesting to be discovered about this.. I'm not sure I understand the distinction you are trying to make. A closed set is always bounded (since it is defined to be a set which includes its own bound). An open set might be bounded or not (open set is one which does not include its own bound - said bound may not exist). A bound is the locus of points whose neighborhoods contain points both within an without the set of interest. So a set is bounded if any such points exist. For the universe, I would guess the exists a bound iff (if and only if) there exist points 'outside' the universe. The question of whether the universe is finite or infinite is implicitly one of extent (even a small box contains an infinite number of points and is therefore an infinite set, so the question is whether the universe is an object of infinite dimensions). If the universe is infinite, then it is probably unbounded and open (unless you are willing to discard the isotropic model of the universe). The interesting stuff starts to happen when you assume a finite universe. If the universe is finite in extent, then there are two choices: bounded and unbounded. In the case that the finite universe is unbounded, then it is probably a three dimensional manifold in a four dimensional space which is analogous to the two dimensional surface of a sphere which is 'closed' in the third dimension (this is the other meaning of 'closed' that the original posting could have meant). Note that there are no bounds on a sphere, no point on the sphere has a neighborhood which contains points outside the sphere (not a two dimensional neighborhood anyway). There would be no bound on the three dimensional analogue of the sphere either. Those people who theorize that the universe is the inside of an enormous black hole are proposing a model of a finite-bounded universe. In this case, the universe is open since a black hole doesn't contain its boundary (the event horizon). What would be beyond the universe is pure speculation, maybe other universes, maybe larger universes which contain ours as an ordinary black hole. Hope this answers the desire for discussion on this topic and that it sparks some more. ------------------------------------------------------------------------------ Reentering Introduction Creates Endlessly Rising Canon, After RICERCAR. - Douglas Hofstadter James Giles
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (01/12/85)
> > I would like to hear a discussion of the implications of a closed and/or > > bounded (not the same thing) universe on such issues. I think there is > > something very interesting to be discovered about this.. I apparently need to clarify: I am interested in quantization and conjugate-transform pairs in a closed and/or bounded manifold. To give a simple example, only certain standing waves are possible on the surface of a spherical membrane. I do not care to discuss whether "closed" means "contains all its limit points" or "countains its boundary" or whatever. That is not the thing that interests me about this topic.