# Net (directed set)

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

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 $D$.