Zero dimensional space

 Yes. In both mathematics and geometry, a single point can be considered a space, and it is indeed zero-dimensional.

Here's the intuition:

• A 1-dimensional space needs one coordinate to specify a location (for example, (x) on a line).
  
• A 2-dimensional space needs two coordinates ((x,y)).
  
• A 3-dimensional space needs three ((x,y,z)).
  

A single point requires no coordinates to move within it, because there is nowhere to move. It has:

• no length,
  
• no width,
  
• no height,
  
• and no internal direction.
  

So its dimension is 0.

There is a nice progression:

Dimension	Example	Degrees of Freedom
0	Point	0
1	Line	1
2	Plane	2
3	Ordinary space	3

A useful way to think about dimension is as the number of independent directions available for motion. At a point, there are none.

This connects intriguingly with the discussion we've been having about spacetime. In Minkowski spacetime:

• An event is represented by a single point in four-dimensional spacetime ((t,x,y,z)).
  
• The event itself is 0-dimensional.
  
• The collection of all events forms the 4-dimensional spacetime manifold.
  

So when physicists say that an event (such as a flash of light occurring at a particular place and time) is a point in spacetime, they are treating it as a zero-dimensional object embedded in a four-dimensional space.

There is also a deeper geometric perspective. Just as a line can be thought of as a collection of points, some mathematicians view dimensions recursively:

• a point is a 0-dimensional space,
  
• a line is generated by moving a point,
  
• a plane by moving a line in a new orthogonal direction,
  
• ordinary space by moving a plane in a new orthogonal direction,
  
• spacetime by adding a time-like dimension to space.
  

This is very similar to the analogy you were making earlier about going from a line to a plane by introducing an orthogonal axis. The "dimension below a line" is precisely a point. The question of what would be "below a point" leads into more abstract mathematics, where some people speak informally of the empty set as having dimension (-1). That's not a physical space, but it plays an important role in topology and geometry.

So, in short: yes—a single point is a perfectly valid space, and it is the canonical example of a zero-dimensional space.