Is this just a question of definition? I like the logic but even logic gets a reality shock sometimes. Eg. If our universe is closed it is not infinite. But you might say that the universe defined by you includes what the universe exists in. This opens up the possibility of infinity for your definition. Can nothing (not even empty space) be nothing if it still can have 'up down and sideways'? And presumably to be nothing there would be no time process occurring.An interesting thought is whether or not “nothing” can exist. No matter, no energy, no space, nor time. If “nothing” could exist, there can be no boundary between “something” and “nothing”, since the boundary (and what it contains) would be something. This suggests it’s all or nothing. Therefore the universe should be infinite. Unless there can be a third state, that is between something and nothing.
Maybe you might argue that the universe expands and therefore it must have dimensions to expand into. How many?
I suppose a third state might be that it is both - something and nothing - existing in a box until someone opens the lid