In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set.
Nets directly generalize the concept of a sequence in a metric space.
Nets are primarily used in the fields of analysis and topology, where they are used to characterize many important topological properties that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces).
The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922.
[2][3] The related concept of a filter was developed in 1937 by Henri Cartan.
As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index
A topological vector space (TVS) is called complete if every Cauchy net converges to some point.
Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-normable) topological vector spaces.
Virtually all concepts of topology can be rephrased in the language of nets and limits.
The following set of theorems and lemmas help cement that similarity: A subset
In general, this statement would not be true if the word "net" was replaced by "sequence"; that is, it is necessary to allow for directed sets other than just the natural numbers if
this neighborhood is a member of the directed set whose index we denote
By the proof given in the next section, it is equal to the set of limits of convergent subnets of
Thus the uniqueness of the limit is equivalent to the Hausdorff condition on the space, and indeed this may be taken as the definition.
This result depends on the directedness condition; a set indexed by a general preorder or partial order may have distinct limit points even in a Hausdorff space.
[13] For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.
[13] He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis, while filters are most useful in algebraic topology.
In any case, he shows how the two can be used in combination to prove various theorems in general topology.
It is in this way that nets are generalizations of sequences: rather than being defined on a countable linearly ordered set (
Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences.
Nets generalize the notion of a sequence so that condition 2 reads as follows: With this change, the conditions become equivalent for all maps of topological spaces, including topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point.
Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behavior.
[14] which makes this one instance where (non-sequence) nets must be used because sequences alone can not reach the desired conclusion.
[b] But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences.
However, the axiom of choice might be need to be assumed to conclude that this tuple
is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections
But if every compact space is also Hausdorff, then the so called "Tychonoff's theorem for compact Hausdorff spaces" can be used instead, which is equivalent to the ultrafilter lemma and so strictly weaker than the axiom of choice.
[16][17][18] Some authors work even with more general structures than the real line, like complete lattices.
Limit superior of a net of real numbers has many properties analogous to the case of sequences.
The definition of the value of a Riemann integral can be interpreted as a limit of a net of Riemann sums where the net's directed set is the set of all partitions of the interval of integration, partially ordered by inclusion.