  We introduce an asymmetric distance function, which we call the `left
Hausdorff distance function', on the space of geodesic laminations on a closed
hyperbolic surface of genus at least 2. This distance is an asymmetric version
of the Hausdorff distance between compact subsets of a metric space. We prove a
rigidity result for the action of the extended mapping class group of the
surface on the space of geodesic laminations equipped with the topology induced
from this distance. More specifically, we prove that there is a natural
homomorphism from the extended mapping class group into the group of bijections
of the space of geodesic laminations that preserve left Hausdorff convergence
and that this homomorphism is an isomorphism.
