Let $X$ be a set. A net is a map from a directed set to $X$ . In other words, it is a pair $(A,\gamma)$ where $A$ is a directed set and $\gamma$ is a map from $A$ to $X$ . If $a\in A$ then $\gamma(a)$ is normally written $x_a$ , and then the net is written $(x_a)_{a\in A}$ , or simply $(x_a)$ if the direct set $A$ is understood.

Now suppose $X$ is a topological space, $A$ is a directed set, and $(x_a)_{a\in A}$ is a net. Let $x\in X$ . Then $(x_a)$ is said to converge to $x$ if whenever $U$ is an open neighbourhood of $x$ , there is some $b \in A$ such that $x_a \in U$ whenever $a \geq b$ .

Similarly, $x$ is said to be an accumulation point (or cluster point) of $(x_a)$ if whenever $U$ is an open neighbourhood of $x$ and $b \in A$ there is $a \in A$ such that $a \geq b$ and $x_a \in U$ .

Nets are sometimes called Moore-Smith sequences, in which case convergence of nets may be called Moore-Smith convergence.

If $B$ is another directed set, and $\delta\colon B\rightarrow A$ is an increasing map such that $\delta(B)$ is cofinal in $A$ , then the pair $(B, \gamma\circ\delta)$ is said to be a subnet of $(A,\gamma)$ . Alternatively, a subnet of a net $(x_\alpha)_{\alpha\in A}$ is sometimes defined to be a net $(x_{\alpha_\beta})_{\beta\in B}$ such that for each $\alpha_0\in A$ there exists a $\beta_0\in B$ such that $\alpha_\beta\geq\alpha_0$ for all $\beta\geq\beta_0$ .

Nets are a generalisation of sequences, and in many respects they work better in arbitrary topological spaces than sequences do. For example:

• If $X$ is Hausdorff then any net in $X$ converges to at most one point.
• If $Y$ is a subspace of $X$ then $x\in\overline{Y}$ if and only if there is a net in $Y$ converging to $x$ .
• if $X'$ is another topological space and $f\colon X\rightarrow X'$ is a map, then $f$ is continuous at $x$ if and only if whenever $(x_a)$ is a net converging to $x$ , $(f(x_a))$ is a net converging to $f(x)$ .
• $X$ is compact if and only if every net has a convergent subnet.