What structure is this definition for projective plane?

Question

I've seen the following definition (or similar) of a projective plane in a few places:

For some field $F$, the projective plane $\mathbb{P}^2(F)$ is the set of all lines in the vector space $F^3$ which pass through the origin.

Here's an example of this on nLab.

However this "definition" seems lacking to me. This just defines a set, with seemingly no additional structure. This surely doesn't seem like the intention, it seems like they would like to imply some additional structure on this set having to do with the properties of the lines in relation to the vector space. For example there could be an implied topology, or metric space.

Now I've picked this particular definition because I've seen it elsewhere, but you might notice all the analytic definitions on nLab define sets with no additional structure. I'm sort of at a loss for what exactly an "Analytic projective plane" even is.

So, what is the intended structure here?

Answer 1

The analytic projective planes have the same structure as the synthetic projective planes. That is they require 3 things

1. A set of points
2. A set of lines
3. An incidence relation between points and lines

And indeed the definition as phrased only provides 1. That is the set of points is all the lines passing through the origin in $F^3$.

The lines are the set of planes passing through the origin in $F^3$ and a point is incident on a line if it is a subset of that plane.

This altogether makes a projective plane.