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Concept
Subnet (mathematics)
Summary
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 and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970.
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet" but they are each 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 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 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.
This article discusses the definition due to Stephen Willard (the other definitions are described in the article Filters in topology#Subnets).
Filters in topology#Subnets
There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard, which is as follows:
If and are nets in a set from directed sets and respectively, then is said to be a of ( or a ) if there exists a monotone final function
such that
A function is , , and an if whenever then and it is called if its is cofinal in
The set being in means that for every there exists some such that that is, for every there exists an such that
Since the net is the function and the net is the function the defining condition may be written more succinctly and cleanly as either or where denotes function composition and is just notation for the function
Importantly, a subnet is not merely the restriction of a net to a directed subset of its domain
In contrast, by definition, a 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.
Official source
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