In metric spaces convergence of sequences unique determines which subsets are closed/open. (A subset $C$ of a metric space is closed if and only if every convergent sequence with all terms in $C$ has limit also in $C$. A subset $O$ of a metric space is open if and only if every convergent sequence which has limit in $O$ is eventually in $O$.) This is no longer true if we work with arbitrary topological spaces, but if we replace sequences by nets, the above characterizations of closed and open sets are valid. This means that convergent nets uniquely determine topology of the space.