Net (directed set)

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