PlanetMath: net

(more info)

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net

(Definition)

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

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

Similarly, is said to be an accumulation point of if whenever is an open neighbourhood of 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.

Now let be another directed set, and let $\delta\colon B\rightarrow A$ be an increasing map such that $\delta(B)$ is cofinal in . Then the pair $(B, \gamma\circ\delta)$ is said to be a subnet of $(A,\gamma)$ .

Under these definitions, nets become a generalisation of sequences to arbitrary topological spaces. For example:

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