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Higher dimensional infinity

If you split a coordinate grid in half, you'll have 2 areas with the same cardinality of infinity, and yet share a single 2-dimensional plane.

So no.
 
If there is a dimension which can contain multiple infinite in size objects, does it mean it's one dimension above them?
Can be either, needs more context
Like how does the separation between the infinite sized objects work?
Is it like extending to different sides from the center towards infinity? Or something else?
 
Can be either, needs more context
Like how does the separation between the infinite sized objects work?
Is it like extending to different sides from the center towards infinity? Or something else?
Multiple 4D structures which expands infinitely in every direction, but never will touch each other. I heard something like that requires an additional spatial axis.
 
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