net

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 (http://planetmath.org/Sequence), and in many respects they work better in arbitrary topological spaces than sequences do. For example:

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

Title	net
Canonical name	Net
Date of creation	2013-03-22 12:54:03
Last modified on	2013-03-22 12:54:03
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	12
Author	yark (2760)
Entry type	Definition
Classification	msc 54A20
Synonym	Moore-Smith sequence
Related topic	Filter
Related topic	NetsAndClosuresOfSubspaces
Related topic	ContinuityAndConvergentNets
Related topic	CompactnessAndConvergentSubnets
Related topic	AccumulationPointsAndConvergentSubnets
Related topic	TestingForContinuityViaNets
Defines	subnet
Defines	Moore-Smith convergence
Defines	cluster point