A net is a generalization of a sequence where a directed set is used as the index set instead of positive integers. Convergence of nets can be defined in a similar way as convergence of sequences. Convergent nets in a topological space uniquely determine its topology.
A net is a generalization of a sequence where a directed set is used as the index set instead of positive integers. Convergence of nets can be defined in a similar way as convergence of sequences.
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.
Also many important topological properties can be characterized using nets (for example Hausdorffness, compactness).
