What is a family of lines?

Question

And furthermore, how do we define the dimension of the family? My question derives from the stating of the Wolff axiom - Let $\mathcal{L}$ be a two-dimensional family of lines in $\mathbb{R}^3$ such that no (affine) plane contains more than a one-dimensional family of lines in $\mathcal{L}$.

Answer 1

The affine group $GL(3, \Bbb R) \ltimes \Bbb R^3$ acts transitively on the space $X$ of lines in $\Bbb R^3$, and so we may view $X$ as a homogeneous space $(GL(3, \Bbb R) \ltimes \Bbb R^3) / H$, where $H$ is the stabilizer of some line in $\Bbb R^3$, which endows $X$ with the structure of a smooth manifold. If a subset $\mathcal{L} \subseteq X$ is, e.g., an embedded submanifold of $X$, we can declare its dimension to be its dimension as a manifold.