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 sequences are unable to characterize. 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 is a non-empty set together with a preorder, typically automatically assumed to be denoted by , with the property that it is also , which means that for any there exists some such that andIn words, this property means that given any two elements, there is always some element that is "above" both of them ; 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 the greatest element. In this case, the conditions and cannot be replaced by the strict inequalities and, since the strict inequalities cannot be satisfied if a or b is the greatest element.
A net in, denoted, is a function of the form whose domain is some directed set, and whose values are. Elements of a net's domain are called its. When the set is clear from context it is simply called a net, and one assumes is a directed set with preorder Notation for nets varies, for example using angled brackets. As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index.
Limits of nets
A net is said to be or a set if there exists some such that for every with the point A point is called a or of the net in whenever:expressed equivalently as: the net or ; and variously denoted as:If is clear from context, it may be omitted from the notation.
If and this limit is unique then one writes:using the equal sign in place of the arrow In a Hausdorff space, every net has at most one limit, and the limit of a convergent net is always unique.
Some authors do not distinguish between the notations and, but this can lead to ambiguities if the ambient space is not Hausdorff.
Cluster points of nets
A net is said to be or if for every there exists some such that and A point is said to be an or cluster point of a net if for every neighborhood of the net is frequently/cofinally in In fact, is a cluster point if and only if it has a subnet that converges to The set of all cluster points of in is equal to for each, where.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, which is as follows:If and are nets then is called a or of if there exists an order-preserving map such that is a cofinal subset of and
The map is called and an if whenever then
The set being in means that for every there exists some such that
If is a cluster point of some subnet of then is also a cluster point of
Ultranets
A net in set is called a or an if for every subset is eventually in or is eventually in the complementEvery constant net is a 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.
If is an ultranet in and is a function then is an ultranet in
Given an ultranet clusters at if and only if it converges to
Cauchy nets
A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.A net is a if for every entourage there exists such that for all is a member of More generally, in a Cauchy space, a net is Cauchy if the filter generated by the net is a Cauchy filter.
A topological vector space 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 if and only if every Cauchy sequence 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 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
A subset is closed in if and only if every limit point in of a net in necessarily lies in.Explicitly, this means that if is a net with for all, and in then
More generally, if is any subset, the closure of is the set of points with for some net in.
Open sets and characterizations of topologies
A subset is open if and only if no net in converges to a point of Also, subset is open if and only if every net converging to an element of is eventually contained inIt 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
A function between topological spaces is continuous at a point if and only if for every net in the domain, in implies inBriefly, a function is continuous if and only if in implies in
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 is not a first-countable space.
Let be continuous at point and let be a net such that
Then for every open neighborhood of its preimage under is a neighborhood of .
Thus the interior of which is denoted by is an open neighborhood of and consequently is eventually in Therefore is eventually in and thus also eventually in which is a subset of Thus and this direction is proven.
Let be a point such that for every net such that Now suppose that is not continuous at
Then there is a neighborhood of whose preimage under is not a neighborhood of Because necessarily Now the set of open neighborhoods of with the containment preorder is a directed set.
We construct a net such that for every open neighborhood of whose index is is a point in this neighborhood that is not in ; that there is always such a point follows from the fact that no open neighborhood of is included in .
It follows that is not in
Now, for every open neighborhood of this neighborhood is a member of the directed set whose index we denote For every the member of the directed set whose index is is contained within ; therefore Thus and by our assumption
But is an open neighborhood of and thus is eventually in and therefore also in in contradiction to not being in for every
This is a contradiction so must be continuous at This completes the proof.
Compactness
A space is compact if and only if every net in has a subnet with a limit in This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.First, suppose that is compact. We will need the following observation. Let be any non-empty set and be a collection of closed subsets of such that for each finite Then as well. Otherwise, would be an open cover for with no finite subcover contrary to the compactness of
Let be a net in directed by For every define
The collection has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that
and this is precisely the set of cluster points of By the proof given in the next section, it is equal to the set of limits of convergent subnets of Thus has a convergent subnet.
Conversely, suppose that every net in has a convergent subnet. For the sake of contradiction, let be an open cover of with no finite subcover. Consider Observe that is a directed set under inclusion and for each there exists an such that for all Consider the net This net cannot have a convergent subnet, because for each there exists such that is a neighbourhood of ; however, for all we have that 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 subnets.Let be a net in a topological space and also let If is a limit of a subnet of then is a cluster point of
Conversely, assume that is a cluster point of
Let be the set of pairs where is an open neighborhood of in and is such that
The map mapping to is then cofinal.
Moreover, giving the product order makes it a directed set, and the net defined by converges to
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.