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 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 ${\displaystyle X=\mathbb {N} }$ 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 Willard (the other definitions are described in the article Filters in topology#Non–equivalence of subnets and subordinate filters).