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[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
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 Willard (the other definitions are described in the article Filters in topology#Non–equivalence of subnets and subordinate filters).
There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,[1] which is as follows: If
(in the sense of Willard or a Willard–subnet[1]) if there exists a monotone final function
is monotone, order-preserving, and an order homomorphism if whenever
Importantly, a subnet is not merely the restriction of a net
In contrast, by definition, a subsequence 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.
if there exists a strictly increasing sequence of positive integers
if and only if there exists a strictly increasing function
is an order-preserving map whose image is cofinal in its codomain and satisfies
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.
is an order-preserving map (because it is a non-decreasing function) whose image
Subnets induced by subsets Suppose
in turn induces a subsequence
induces a canonical subnet that may be written as a subsequence.
However, as demonstrated below, not every subnet of a sequence is a subsequence.
The definition generalizes some key theorems about subsequences: Taking
be the identity map in the definition of "subnet" and requiring
leads to the concept of a cofinal subnet, 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.
In other words, every cluster point of a net in a subset belongs to the closure of that set.
is necessarily a cluster point of that net.
is a cluster point of a given net if and only if it has a subnet that converges to