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Let's say someone who is in a part of the universe that appears very far from us has their own deep spacecraft and sends it out for a look at the universe.

Would they see the same galaxies and stuff that we can, but from different angles, or would they see other light sources that we can't because they're too far from us, and not be able to see some of the stuff that we can because it's too far out?

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Well, this depends obviously how far they are away. If their observable universe and ours overlap, then objects in that overlap will be observable by us and them. However, we will see them from different orientation and (for most objects) at a different redshift.

Here, observable universe is the part of the universe which is close enough for light emitted to reach the observer now, i.e. is roughly of the radius $c/H$.

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