Net (mathematics) on arvau

Wikipedia / Net (mathematics)

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. The codomain of this function is usually some topological space. 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). Nets are in one-to-one correspondence with filters.

History

The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922. The term "net" was coined by John L. Kelley.

The related concept of a filter was developed in 1937 by Henri Cartan.

Definitions

A directed set together with a preorder (unless indicated otherwise), with the property that it is also () ), there is always some element that is "above" both of them (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are required to be total orders or even partial orders. A directed set may have a greatest element, since the strict inequalities cannot be satisfied if a or b is a greatest element.

A net in, is a function whose domain. Elements of a net's domain are called its is clear from context it is simply called a net. As is common in algebraic topology.

Limits of nets

is said to be or is called a or whenever:

for every open neighborhood,

expressed equivalently as: the net or is clear from context, it may be omitted from the notation.

In a Hausdorff space, the limit of a convergent net is unique., but this can lead to ambiguities if the ambient space is not Hausdorff.

Cluster points of nets

is said to be or is said to be an or cluster point.

Subnets

The analogue of "subsequence" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard, is called a or is a cofinal is called and an being

Ultranets

is called a or an

Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet. Assuming the axiom of choice, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.

Cauchy nets

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.

is a if for every entourage More generally, in a Cauchy space is Cauchy if the filter generated by the net is a Cauchy filter.

A topological vector space (TVS) is called if every Cauchy net converges to some point. A normed space, which is a special type of topological vector space, is a complete TVS (equivalently, a Banach space) if and only if every Cauchy sequence converges to some point (a property that is called ). 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.

Characterizations of topological properties

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. The following set of theorems and lemmas help cement that similarity:

Closed sets and closure

is any subset, the closure.

Open sets and characterizations of topologies

It is these characterizations of "open subset" that allow nets to characterize topologies. Topologies can also be characterized by closed subsets since a set is open if and only if its complement is closed. So the characterizations of "closed set" in terms of nets can also be used to characterize topologies.

Continuity

between topological spaces is continuous is not a first-countable space (or not a sequential space).

). Thus the interior and this direction is proven.

Then there is a neighborhood with the containment preorder is a directed set as well).

This completes the proof.

Compactness

is compact This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.

is compact. We will need the following observation (see finite intersection property

has a convergent subnet.

This is a contradiction and completes the proof.

Cluster and limit points

The set of cluster points of a net is equal to the set of limits of its convergent subnetss.

the product order

A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.

Other properties

is a Hausdorff space with two distinct limits. Thus the uniqueness of the limit is 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.

Relation to filters

A filter is a related idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence. More specifically, every filter base induces an generated by this filter base is called the net's . Convergence of the net implies convergence of the eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other. 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.

Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts. 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.

The learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especially analysts, prefer them over filters. However, filters, and especially ultrafilters, have some important technical advantages over nets that ultimately result in nets being encountered much less often than filters outside of the fields of analysis and topology.

As generalization of sequences

Every non-empty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers together with the usual integer Conversely, any net whose domain is the natural numbers is a sequence It is in this way that nets are generalizations of sequences: rather than being defined on a countable linearly ordered), a net is defined on an arbitrary directed set is taken from sequences.

Similarly, every limit of a sequence and limit of a function contains infinitely many elements of the sequence.

While condition 1 always guarantees condition 2, the converse is not necessarily true. The spaces for which the two conditions are equivalent are called sequential spaces. All first-countable spaces, including metric spaces, are sequential spaces, but not all topological spaces are sequential. Nets generalize the notion of a sequence so that condition 2 reads as follows:

(continuous in the net sense).

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.

with the tuple and conversely) and endow it with the product topology is identical to the topology of pointwise convergence However, there does not exist any This partial order turns the identity map

More generally, a subnet of a sequence is necessarily a sequence. Moreso, a subnet of a sequence may be a sequence, but not a subsequence..

Examples

Subspace topology

is endowed with the subspace topology depends and the image

Neighborhood systems

denote the set of all neighbourhoods if and only if Therefore, in this neighborhood system according to the definition of net convergence.

Given a subbase (where note that every base This characterization extends to neighborhood subbases (and so also neighborhood bases

Limits in a Cartesian product

A net in the product space has a limit if and only if each projection has a limit.

be topological spaces, endow their Cartesian product with the product topology

It is sometimes useful to think of this definition in terms of function composition

However, the converse does not hold in general. since the open ball

Tychonoff's theorem and relation to the axiom of choice

However, the axiom of choice is the is a Hausdorff space are surjective maps.

The axiom of choice is equivalent to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. 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. Nets can be used to give short proofs of both version of Tychonoff's theorem by using the characterization of net convergence given above together with the fact that a space is compact if and only if every net has a convergent subnet.

Limit superior/inferior

Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences. Some authors work even with more general structures than the real line, like complete lattices.

where equality holds whenever one of the nets is convergent.

Riemann integral

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.

Metric spaces

is a metric space (or a pseudometric space is endowed with the metric topology is a net of real numbers. In plain English is a normed space (or a seminormed space