Subnet (mathematics)
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 and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970. Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet" but they are each  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  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. This article discusses the definition due to Willard (the other definitions are described in the article Filters in topology#Subnets). Definitions

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 in a set  from directed sets  and  respectively, then  is said to be a  of  ( or a ) if there exists a monotone final function

such that 
 
A function  is , , and an  if whenever  then  and it is called  if its image  is cofinal in  
The set  being  in  means that for every  there exists some  such that  that is, for every  there exists an  such that 

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

Importantly, a subnet is not merely the restriction of a net  to a directed subset of its domain  
In contrast, by definition, a  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  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 

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. Subnet of a sequence that is not a sequence

A subnet of a sequence is  necessarily a sequence. 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

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 , 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

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

where  for each

Convergence versus clustering

If a net converges to a point  then  is necessarily a cluster point of that net. The converse is not guaranteed in general. That is, it is possible for  to be a cluster point of a net  but for  to  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

See also

Notes

Citations

References

 
 
 
  
  

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