  The signature of a $p$-weakly geometric rough path summarises a path up to a
generalised notion of reparameterisation. The quotient space of equivalence
classes on which the signature is constant yields unparameterised path space.
The study of topologies on unparameterised path space, initiated in [CT24b] for
paths of bounded variation, has practical bearing on the use of signature based
methods in a variety applications. This note extends the majority of results
from [CT24b] to unparameterised weakly geometric rough path space. We study
three classes of topologies: metrisable topologies for which the quotient map
is continuous; the quotient topology derived from the underlying path space;
and an explicit metric between the tree-reduced representatives of each
equivalence class. We prove that topologies of the first type (under an
additional assumption) are separable and Lusin, but not locally compact or
completely metrisable. The quotient topology is Hausdorff but not metrisable,
while the metric generating the third topology is not complete and its topology
is not locally compact. We also show that the third topology is Polish when
$p=1$.
