I'm new in the field, so I'm sorry in advance if my question is too naive.

Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed geodesics on $S$ with respect to $g$.

Usually on $\mathcal{S}(S)$ there is the topology induced by the hausdorff metric (indentifying a geodesic with its image, we can view $\mathcal{S}(S)$ as a subset of $\mathcal{C}(S)$, the set of closed sets of $S$).

Lately I've stumbled upon the "$\textit{compact-open topology}$" on $\mathcal{S}(S)$: this topology has as a subbase the sets $V(K,U)$ of isometries $h:K\rightarrow U$, where $K$ is a compact subset of $\mathbb{R}$ and $U$ is an open set in $S$.

My questions are:

**1) Why one should use the compact-open topology instead of the topology induced by the hausdorff metric on $\mathcal{S}(S)$?** Maybe it has something to do with conververgence of sequences of geodesics?

**2) When is a subset $V\subset\mathcal{S}(S)$ closed for the compact open topology?**

**3) Can you point me out some references to study properties of the compact open topology on $\mathcal{S}(S)$?**

Thank you

parameterized geodesicswhile the Hausdorff topology is on the space of unparameterized geodesics. Of course, there is a map from one to the other, but the sets are different. One also frequently uses the Hausdorff topology on the set of all closed subsets of $S$ so that the limit of a sequence of unparameterized geodesics is not a geodesic but a different object, say, a geodesic lamination in the context of simple geodesics. $\endgroup$ – Misha Nov 20 '15 at 19:59