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.

If (xα) and (yβ) are nets from directed sets A and B respectively, then (yβ) is a subnet of (xα) if there exists a monotone cofinal function
 * h : B &rarr; A

such that
 * yβ = xh(β).

A function h : B &rarr; A is monotone if β1 &le; β2 implies h(β1) &le; h(β2) and cofinal if its image is cofinal in A&mdash;that is, for every α in A there exists a β in B such that h(β) ≥ α.

While complicated, the definition does generalize some key theorems about subsequences:


 * A net (xα) converges to x if and only if every subnet of (xα) coverges to x.
 * A net (xα) has a cluster point y if and only if it has a subnet (yβ) that converges to y.
 * A topological space X is compact if and only if every net in X has a convergent subnet (see Net_(mathematics) for a proof).

A 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.

Note that 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. 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.