I guess all the stars and planets would be in the same place (i.e. since they couldn’t be separated by space), so they would all merge together and form a giant black hole. And that would be the end of the universe.
If there was no space we wouldn’t be here to ask that question, for a quite subtle reason:
We think that the Universe started in a hot Big Bang 13.7 billion years ago, and expanded and cooled down ever since. So space is a consequence of the universe expanding after the Big Bang. It took so long for us as human being to be here because it takes time to form the first stars; those stars then die, and as they do so produce heavy elements such as iron, carbon and oxygen that are needed for life. Then a rocky planet like the Earth can form. Finally, you need to wait another few billion years before life can appear and evolve.
So you see that a universe where there is life like us needs to be very old. And if it is old, it needs to be big — that’s because the Universe expands and becomes bigger over time.
So if there as no space, that would mean that we would be at a very early time in the history of the universe, and that’s impossible because the conditions for life are not there yet!
So it’s a very deep question — the existence of space (a lot of it!) is actually required for us to be here in the first place and ask that (very clever) question!
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