# Net (directed set)

From Encyclopedia of Mathematics

A mapping of a directed set into a (topological) space.

#### Comments

The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net in a topological space converges to a point if for each open neighbourhood of in the net is eventually in . The last phrase means that there is an such that for all in .

The theory of convergence of nets is known as Moore–Smith convergence, [a1].

#### References

[a1] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. Chapt. II |

**How to Cite This Entry:**

Net (directed set). M.I. Voitsekhovskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Net_(directed_set)&oldid=17428

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098