I am unsure if this belongs to math or philosophy.
Let's say there's 0-dimensional space, however multiple objects exist within in, occupying the same "spot". If multiple objects exist, is the space still zero-dimensional?
The problem here is that multiple objects existing at the same spot introduces dimensionality. If objects exist, they are somehow distinct, and can be ordered around.
At the same time, as far as I know, there is no axiom saying that only one point can exist at given coordinates. And in 3d space, each point of space can have phenomenon associated with it, which may require a higher-dimensional construct to represent it. For example, electromagnetic spectrum at a point could be expressed as 1-dimensional line.
This implies that a single point can have a set of properties associated with it, which can be expressed in dimensional way, but that does not affect dimensionality of the space itself, as they do not form a spatial dimension.
Which line of reasoning is correct?
The only zero dimensional space, by definition, is the space containing only one element, the zero element. This means that a zero dimensional space cannot have multiple objects existing within it. All reasoning after we assume the existence of such a space is meaningless.