Subnet (mathematics)

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In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.

There are three non-equivalent definitions of "subnet". The first definition of a subnet was introduced by John L. Kelley in 1955[1] and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970.[1] Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet"[1] but they are each not equivalent to the concept of "subordinate filter", which is the analog of "subsequence" for filters (they are not equivalent in the sense that there exist subordinate filters on whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship). A third definition of "subnet" (not equivalent to those given by Kelley or Willard) that is equivalent to the concept of "subordinate filter" was introduced independently by Smiley (1957), Aarnes and Andenaes (1972), Murdeshwar (1983), and possibly others, although it is not often used.[1]

This article discusses the definition due to Willard (the other definitions are described in the article Filters in topology#Non–equivalence of subnets and subordinate filters).

Definitions

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There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,[1] which is as follows: If   and   are nets in a set   from directed sets   and   respectively, then   is said to be a subnet of   (in the sense of Willard or a Willard–subnet[1]) if there exists a monotone final function   such that   A function   is monotone, order-preserving, and an order homomorphism if whenever   then   and it is called final if its image   is cofinal in   The set   being cofinal in   means that for every   there exists some   such that   that is, for every   there exists an   such that  [note 1]

Since the net   is the function   and the net   is the function   the defining condition   may be written more succinctly and cleanly as either   or   where   denotes function composition and   is just notation for the function  

Subnets versus subsequences

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Importantly, a subnet is not merely the restriction of a net   to a directed subset of its domain   In contrast, by definition, a subsequence of a given sequence   is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. Explicitly, a sequence   is said to be a subsequence of   if there exists a strictly increasing sequence of positive integers   such that   for every   (that is to say, such that  ). The sequence   can be canonically identified with the function   defined by   Thus a sequence   is a subsequence of   if and only if there exists a strictly increasing function   such that  

Subsequences are subnets

Every subsequence is a subnet because if   is a subsequence of   then the map   defined by   is an order-preserving map whose image is cofinal in its codomain and satisfies   for all  

Sequence and subnet but not a subsequence

The sequence   is not a subsequence of   although it is a subnet because the map   defined by   is an order-preserving map whose image is   and satisfies   for all  [note 2]

While a sequence is a net, a sequence has subnets that are not subsequences. The key difference is that subnets can use the same point in the net multiple times and the indexing set of the subnet can have much larger cardinality. Using the more general definition where we do not require monotonicity, a sequence is a subnet of a given sequence, if and only if it can be obtained from some subsequence by repeating its terms and reordering them.[2]

Subnet of a sequence that is not a sequence

A subnet of a sequence is not necessarily a sequence.[3] For an example, let   be directed by the usual order   and define   by letting   be the ceiling of   Then   is an order-preserving map (because it is a non-decreasing function) whose image   is a cofinal subset of its codomain. Let   be any sequence (such as a constant sequence, for instance) and let   for every   (in other words, let  ). This net   is not a sequence since its domain   is an uncountable set. However,   is a subnet of the sequence   since (by definition)   holds for every   Thus   is a subnet of   that is not a sequence.

Furthermore, the sequence   is also a subnet of   since the inclusion map   (that sends  ) is an order-preserving map whose image   is a cofinal subset of its codomain and   holds for all   Thus   and   are (simultaneously) subnets of each another.

Subnets induced by subsets

Suppose   is an infinite set and   is a sequence. Then   is a net on   that is also a subnet of   (take   to be the inclusion map  ). This subnet   in turn induces a subsequence   by defining   as the   smallest value in   (that is, let   and let   for every integer  ). In this way, every infinite subset of   induces a canonical subnet that may be written as a subsequence. However, as demonstrated below, not every subnet of a sequence is a subsequence.

Applications

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The definition generalizes some key theorems about subsequences:

  • A net   converges to   if and only if every subnet of   converges to  
  • A net   has a cluster point   if and only if it has a subnet   that converges to  
  • A topological space   is compact if and only if every net in   has a convergent subnet (see net for a proof).

Taking   be the identity map in the definition of "subnet" and requiring   to be a cofinal subset of   leads to the concept of a cofinal subnet, which turns out to be inadequate since, for example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.

Clustering and closure

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If   is a net in a subset   and if   is a cluster point of   then   In other words, every cluster point of a net in a subset belongs to the closure of that set.

If   is a net in   then the set of all cluster points of   in   is equal to[3]   where   for each  

Convergence versus clustering

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If a net converges to a point   then   is necessarily a cluster point of that net.[3] The converse is not guaranteed in general. That is, it is possible for   to be a cluster point of a net   but for   to not converge to   However, if   clusters at   then there exists a subnet of   that converges to   This subnet can be explicitly constructed from   and the neighborhood filter   at   as follows: make   into a directed set by declaring that   then   and   is a subnet of   since the map   is a monotone function whose image   is a cofinal subset of   and  

Thus, a point   is a cluster point of a given net if and only if it has a subnet that converges to  [3]

See also

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Notes

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  1. ^ Some authors use a more general definition of a subnet. In this definition, the map   is required to satisfy the condition: For every   there exists a   such that   whenever   Such a map is final but not necessarily monotone.
  2. ^ Indeed, this is because   and   for every   in other words, when considered as functions on   the sequence   is just the identity map on   while  

Citations

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  1. ^ a b c d e f Schechter 1996, pp. 157–168.
  2. ^ Gähler, Werner (1977). Grundstrukturen der Analysis I. Akademie-Verlag, Berlin., Satz 2.8.3, p. 81
  3. ^ a b c d Willard 2004, pp. 73–77.

References

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