Weak topologies for unbounded nets in CAT($0$) spaces

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.