# 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 $ S : D \rightarrow X $ in a topological space $ X $ converges to a point $ s \in X $ if for each open neighbourhood $ U $ of $ s $ in $ X $ the net $ S $ is eventually in $ U $. The last phrase means that there is an $ m \in D $ such that $ S ( n) \in U $ for all $ n \geq m $ in $ D $.

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).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Net_(directed_set)&oldid=47955

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article