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.
If and are nets from directed sets A and B respectively, then is a subnet of if there exists a monotone final function
such that
A function h : B → A is monotone if β1 ≤ β2 implies h ≤ h and final if its image is cofinal in A—that is, for every α in A there exists a β in B such that h ≥ α.
While complicated, the definition does generalize some key theorems about subsequences:
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 is a subnet of the net. 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.