Net (mathematics)

A net is a function from a directed set into a set, usually a topological space. It generalizes a sequence by replacing the natural-number index set with an index set that need not be countable or linearly ordered. Nets allow convergence to describe closure, continuity, and compactness in arbitrary spaces, where sequences alone may be insufficient. (math.wvu.edu)

Definition and indexing

A directed set is a nonempty set $D$ equipped with a reflexive, transitive relation $\leq$ such that any two indices have a common upper bound:

Antisymmetry is sometimes required, making the relation a partial order; allowing a preorder also gives the usual theory of nets. A net in $X$ is a map

written $(x_d)_{d\in D}$. An ordinary sequence is the case $D=\mathbb N$ with its usual order. (math.wvu.edu)

The order indicates what it means to proceed sufficiently far through the indices. Different indices need not be comparable, but directedness ensures that finitely many requirements on how far to proceed can be satisfied simultaneously. The net is the indexed function, not merely its range: its indexing determines its eventual behavior. (math.wvu.edu)

Typical directed sets include the finite subsets of a set ordered by inclusion, and the neighborhoods of a point ordered by reverse inclusion. For neighborhoods, smaller sets count as later indices; intersections provide common upper bounds. (arxiv.org)

Convergence and cluster points

In a topological space $X$, a net $(x_d)$ converges to $x$, written $x_d\to x$, if for every neighborhood $U$ of $x$, there is an index $d_0$ such that

Thus the net eventually remains in every neighborhood of its limit. This requires neither a distance function nor countable indexing. (math.uchicago.edu)

A net is eventually in a set $A$ if all its terms beyond some index belong to $A$. It is frequently in $A$ if

A point $x$ is a cluster point of the net if the net is frequently in every neighborhood of $x$. Being a cluster point is weaker than being a limit: arbitrarily late visits do not require eventual residence. (math.wvu.edu)

Limits need not be unique. A space is a Hausdorff space if and only if every net has at most one limit. In a Hausdorff space, disjoint neighborhoods of two proposed distinct limits would eventually have to contain the same terms, which is impossible. (lerman.web.illinois.edu)

Why sequences are insufficient

In a first-countable space, every point has a countable neighborhood base, and membership in the closure of a set can be witnessed by a convergent sequence from that set. This includes every metric space. Arbitrary topological spaces need not have this property. (arxiv.org)

For example, let $\omega_1$ be the first uncountable ordinal, and give $[0,\omega_1]$ its order topology. The point $\omega_1$ belongs to the closure of $[0,\omega_1)$, but no sequence of countable ordinals converges to it: the countably many terms have an upper bound strictly below $\omega_1$. The ordinal-indexed net

does converge to $\omega_1$. It eventually exceeds every prescribed countable ordinal. This is a concrete instance of the uncountable-ordinal phenomenon underlying standard examples such as the extended long line. (math.uchicago.edu)

The distinction is not simply between countable and uncountable lists. Nets also permit partially ordered indexing, which accommodates several independent approximation requirements. (arxiv.org)

Subnets

A subnet generalizes a subsequence, but its index set may differ substantially from that of the original net. Under a common convention, a subnet of $(x_d)_{d\in D}$ is a net

where $E$ is directed and $\phi:E\to D$ is order-preserving and cofinal. Cofinality means that for every $d\in D$, some $\phi(e)$ satisfies $\phi(e)\geq d$. Together, these requirements ensure that the subnet eventually passes beyond every original index. (legacy-www.math.harvard.edu)

Other conventions replace order preservation and cofinality with the direct requirement

Authors must specify which convention they use. A subnet need not merely restrict the original net to a subset of its indices, and a subnet of a sequence need not be a subsequence. (math.wvu.edu)

Every subnet of a convergent net converges to the same limit. A point is a cluster point of a net if and only if some subnet converges to it. (math.hu-berlin.de)

Characterizations of topological properties

Nets provide several general characterizations:

• Closure: $x$ belongs to the closure of $A\subseteq X$ if and only if a net with all its terms in $A$ converges to $x$.
• Closed sets: $A$ is closed if and only if it contains every limit, in $X$, of every net in $A$.
• Continuity: A map $f:X\to Y$ is a continuous function if and only if $x_d\to x$ always implies $f(x_d)\to f(x)$.
• Products: A net in a space with the product topology converges if and only if it converges in each coordinate. (math.wvu.edu)

For closure, one can construct a witnessing net explicitly. If $x\in\overline A$, each neighborhood $U$ of $x$ meets $A$. Choose $a_U\in A\cap U$, and index these points by neighborhoods ordered by reverse inclusion. Then $a_U\to x$. This explains how nets follow the full neighborhood structure rather than a potentially inadequate countable selection. (arxiv.org)

A space is compact if and only if every net in it has a convergent subnet; equivalently, every net has a cluster point. This is the net counterpart of sequential compactness, but the two notions are not equivalent in arbitrary spaces. They coincide in metric spaces. (math.hu-berlin.de)

Relationship with filters

A filter describes eventual behavior through subsets rather than indices. For a net $(x_d)$, define its tail sets by

These generate the filter

Thus $\mathcal F_x$ consists precisely of the sets in which the net is eventually located. The net converges to $x$ exactly when every neighborhood of $x$ belongs to $\mathcal F_x$. (math.wvu.edu)

Conversely, every proper filter can be represented by a net with the same eventual-set filter. One construction uses pairs $(A,a)$, where $A$ belongs to the filter and $a\in A$, orders the pairs by reverse inclusion of their set components, and assigns the value $a$. Nets and filters therefore express equivalent convergence information, but they are not in a literal one-to-one correspondence: different nets can generate the same filter. (math.wvu.edu)

Historical development

The general theory of directed-index convergence was developed by E. H. Moore and H. L. Smith in “A General Theory of Limits,” published in the American Journal of Mathematics in April 1922. The alternative name Moore–Smith sequence reflects this origin. Their framework unified limit processes whose indexing was more general than the positive integers. (jstor.org)

References

1. A General Theory of Limitsjstor.org
2. Math 535 lecture notes and videos, Fall 2021lerman.web.illinois.edu
3. Topology I—III, HU Berlinmath.hu-berlin.de
4. Topology Course Notes — Harvard Universitylegacy-www.math.harvard.edu
5. Topology - Imath.uchicago.edu
6. Sequences and nets in topologyarxiv.org