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.

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 a subnet of $x_{\bullet }$ if there exists a monotone final function

h:B\to A

such that

y_{b}=x_{h(b)}

for all

b\in B.

A function $h:B\to A$ is monotone if $b\leq c$ imples $h(b)\leq h(c)$ 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, ...) is a subnet of the net (1, 2, 3, 4, ...). 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 don't 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.^[1]

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

^ 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.
Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796.

[1] 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.

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

[note 1]