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:

• 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 $X^{\prime }$ 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 $f(x)$.

• $X$ 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