# Net (topology)

From Citizendium, the Citizens' Compendium

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 *m* ≥ *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*.

## References

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