  Weak topologies that yield weak convergence for bounded sequences and nets in
CAT($0$) spaces have been studied in the past. We are here concerned with weak
topologies that yield weak convergence of unbounded sequences and nets. We
analyze two such topologies that generalize the weak topology on Hilbert spaces
and that agree with the strong topology on a CAT($0$) space if and only if the
space is locally compact.
