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 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 $X=\mathbb {N}$ 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 Stephen Willard (the other definitions are described in the article Filters in topology#Subnets).

Definitions

If $x_{\bullet }=\left(x_{a}\right)_{a\in A}$ and $y_{\bullet }=\left(y_{b}\right)_{b\in B}$ are nets from directed sets $A$ and $B$ respectively, then $y_{\bullet }$ is said to be a subnet of $x_{\bullet }$ (in the sense of Willard or a Willard–subnet) if there exists a monotone final function

h:B\to A

such that

y_{b}=x_{h(b)}\qquad {\text{ for all }}b\in B.

A function $h:B\to A$ is monotone if $b\leq c$ implies $h(b)\leq h(c)$ and it is called final if its image is cofinal in $A$ —that is, for every $a\in A$ there exists a $b\in B$ such that $h(b)\geq a.$ [note 1]

Applications

The definition generalizes some key theorems about subsequences:

A net $x_{\bullet }$ converges to $x$ if and only if every subnet of $x_{\bullet }$ converges to $x.$
A net $x_{\bullet }$ has a cluster point $y$ if and only if it has a subnet $y_{\bullet }$ that converges to $y$
A topological space $X$ is compact if and only if every net in $X$ has a convergent subnet (see net for a proof).

A seemingly more natural definition of a subnet would be to require $B$ to be a cofinal subset of $A$ and that $h$ be the identity map. This concept, known as a cofinal subnet, turns out to be inadequate. For example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.

While a sequence is a net, a sequence has subnets that are not subsequences. For example the net $(1,1,2,3,4,\ldots )$ is a subnet of the net $(1,2,3,4,\ldots ).$ 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]

Notes

Some authors use a slightly more general definition of a subnet. In this definition, the map $h$ is required to satisfy the condition: For every $a\in A$ there exists a $b_{0}\in B$ such that $h(b)\geq a$ whenever $b\geq b_{0}.$ Such a map is final but not necessarily monotone.

Citations

Schechter 1996, pp. 157–168.
Gähler, Werner (1977). Grundstrukturen der Analysis I. Akademie-Verlag, Berlin., Satz 2.8.3, p. 81

References

Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3885380064.
Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.
Runde, Volker (2005). A Taste of Topology. Springer. ISBN 978-0387-25790-7.
Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
Willard, Stephen (2004) [1970]. General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

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[2] Some authors use a slightly more general definition of a subnet. In this definition, the map $h$ is required to satisfy the condition: For every $a\in A$ there exists a $b_{0}\in B$ such that $h(b)\geq a$ whenever $b\geq b_{0}.$ Such a map is final but not necessarily monotone.

[FOOTNOTESchechter1996157–168-1] Schechter 1996, pp. 157–168.

[3] Gähler, Werner (1977). Grundstrukturen der Analysis I. Akademie-Verlag, Berlin., Satz 2.8.3, p. 81