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 $a\ge b$.
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 $a\ge b$ 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 $\delta :B\to A$ 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 ${\alpha}_{\beta}\ge {\alpha}_{0}$ for all $\beta \ge {\beta}_{0}$.
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 $f:X\to {X}^{\prime}$ 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  20130322 12:54:03 
Last modified on  20130322 12:54:03 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  12 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 54A20 
Synonym  MooreSmith sequence 
Related topic  Filter 
Related topic  NetsAndClosuresOfSubspaces 
Related topic  ContinuityAndConvergentNets 
Related topic  CompactnessAndConvergentSubnets 
Related topic  AccumulationPointsAndConvergentSubnets 
Related topic  TestingForContinuityViaNets 
Defines  subnet 
Defines  MooreSmith convergence 
Defines  cluster point 