Identifying the space of geodesics on the hyperbolic plane as a topological space. On pg. 2 of this PDF, the author defines $\mathcal G=(-\infty, 0)\times(0, 1)$ and mentions that $\mathcal G$ can be thought of as "a space of geodesics on the hyperbolic $2$-space $\mathbb H^2$."
One of the things that is confusing me is the use of the indefinite article "a" in the phrase "a space of geodesics..."
The other thing is that how are we thinking of the collection of geodesics as a topological space?
Can somebody please spell out the details here, and please feel free to offer an insights.
 A: What this means, I believe, is that it is a subspace of the full space of geodesics. In the upper half plane model $\mathbb H$, the boundary is 
$$\partial \mathbb H = \mathbb R \cup \{\infty\}
$$
which is topologized as the one point compactification of $\mathbb R$, and is therefore homeomorphic to the circle $S^1$. 
The full space of (oriented) geodesics can be identified with the set of ordered pairs 
$$\{(x,y) \in \partial \mathbb H \times \partial \mathbb H \mid x \ne y \}
$$
where $x$ is the initial ideal endpoint of the geodesic and $y$ is the terminal ideal endpoint. That space can be topologized as a subset of the torus $\partial \mathbb H \times \partial \mathbb H \approx S^1 \times S^1$ with the product topology.
In that passage, the author seems to be thinking of $(-\infty,0) \times (0,1)$ very literally as a subset of $\partial \mathbb H \times \partial \mathbb H$, with the subspace topology --- which, of course, is also the ordinary product topology on $(-\infty,0) \times (0,1)$.
So, the reason that it is "a" space of geodesics is simply because it is "a" subset of the full space of geodesics.
