I am a bit shaky on this answer, but let me give it a try. In a closed universe, space is always bounded in the sense that it has a finite volume at any epoch. In an infinite universe, space is always unbounded, even at the initial singularity, however, in terms of the histories of particles, the initial singularity is the origin of all world lines since along any world line the past history of a particle ends abruptly, not asymptotically, at the Big Bang singularity. In this sense, even in an infinite universe, which by the way has always had an infinite spatial volume even at the Big Bang, the initial Big Bang singularity does provide a boundary to the universe. But this boundary exists in space-time, not simply in the 3-dimensional spatial portion of its geometry. It is important to distinguish between the full, 4- dimensional space-time geometry and the 3-dimensional space-like geometry of this problem.
Copyright 1997 Dr. Sten Odenwald
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