Is earth part of space or is it its own thing?
Earth exists within space.
It depends on your choice of reference. If our reference point is the earth/ground, most people would typically say that space is everything that isn't the earth. If you're a sciency type, your reference point might be "space" or "the universe," so you'd say the earth is just another part of space.
If you mean that Earth is separate from space (like in a small bubble outside space) then no. Earth is indeed in space. "Space" is quite well named, as it's in every space that isn't taken by an object, e.g. a brick. So when you're walking along the road, you're still in space, as to say.
"Space is what you swing a cat in" is a fun reference to the "Schrodinger cat"
So by that definition, Earth is already taking that place.
If you have a three dimensional illustration and earth plopped in a part of it, then you could say earth is part of that space. Or you could decide some of the space is occupied and some is not.
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