I strongly believe Space can't exist everywhere.
What will happen to a Spacecraft trying to leave the Universe and enters a region of no Space?
If a Spacecraft traveled in one direction through the Universe where would it end?
In my opinion, it depends on what you mean by space.
If by space you mean the geometric space, Euclid deals with solid bodies’ forms without studying the space embracing them. Projective geometry, a later formalized synthetic geometry, is a spatial geometry because of the point-plane duality principle. In fact, it deals with spatial configurations. When drawing them, it is never presumed that they end at the edge of the sheet of paper. In analytic geometry, by space is often meant a number field. The properties of the number field are associated with analytic space.
If by space you mean the world with which you interact directly, then you probably may have the experience of an impassable boundary, which however could be overflown, for example. Contrarily, if you look at the sky, you don’t have the perception that it ends somewhere, or even that it occupies a given location. In fact, you only experience it with your sense of sight. I don't know if that experience changes, when looking at it from a spacecraft.
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