Convergence of time scales

The set of time scales is the hyperspace $\mathrm{CL}(\mathbb{R})$. There are three popular topologies on hyperspaces: the induced topology by the Hausdorff metric, the Vietoris topology, and the Fell topology.

Which topology should be used on $\mathrm{CL}(\mathbb{R})$?

Example: Consider the sequence $\mathbb{T}_n = [0,n]$ of time scales. As $n$ approaches $\infty$ one would expect that the time scales to converge to the set $[0,\infty) \subset \mathbb{R}$. This does not occur, because we can compute

References

<bibtex>

@inproceedings{MR2547668,
  title="The Fell topology on the space of time scales for dynamic equations",
  author="Oberste-Vorth, Ralph W.",
  booktitle="Advances in Dynamical Systems and Applications 2008",
}

</bibtex>