Net (topology)

In topology, a net is a function on a directed set into a topological space which generalises the notion of sequence. Convergence of a net may be used to completely characterise the topology.

A directed set is a partially ordered set D in which any two elements have a common upper bound. A net in a topological space X is a function a from a directed set D to X.

A net is eventually in a subset S of X if there is an index n in D such that for all m &ge; n we have a(m) in S.

A net converges to a point x in X if it is eventually in any neighbourhood of x.