Net (topology)

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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.

The natural numbers with the usual order form a directed set, and so a sequence is a special case of a net.

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

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

References

  • J.L. Kelley (1955). General topology. van Nostrand, 62-83.