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 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.

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, which is as follows: If ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ and ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i\in I}}$ are nets in a set ${\displaystyle X}$ from directed sets ${\displaystyle A}$ and ${\displaystyle I,}$ respectively, then ${\displaystyle s_{\bullet }}$ is said to be a subnet of ${\displaystyle x_{\bullet }}$ (in the sense of Willard or a Willard–subnet) if there exists a monotone final function $${\displaystyle h:I\to A}$$ such that $${\displaystyle s_{i}=x_{h(i)}\quad {\text{ for all }}i\in I.}$$ A function ${\displaystyle h:I\to A}$ is monotone, order-preserving, and an order homomorphism if whenever ${\displaystyle i\leq j}$ then ${\displaystyle h(i)\leq h(j)}$ and it is called final if its image ${\displaystyle h(I)}$ is cofinal in ${\displaystyle A.}$ The set ${\displaystyle h(I)}$ being cofinal in ${\displaystyle A}$ means that for every ${\displaystyle a\in A,}$ there exists some ${\displaystyle b\in h(I)}$ such that ${\displaystyle b\geq a;}$ that is, for every ${\displaystyle a\in A}$ there exists an ${\displaystyle i\in I}$ such that ${\displaystyle h(i)\geq a.}$

Since the net ${\displaystyle x_{\bullet }}$ is the function ${\displaystyle x_{\bullet }:A\to X}$ and the net ${\displaystyle s_{\bullet }}$ is the function ${\displaystyle s_{\bullet }:I\to X,}$ the defining condition ${\displaystyle \left(s_{i}\right)_{i\in I}=\left(x_{h(i)}\right)_{i\in I},}$ may be written more succinctly and cleanly as either ${\displaystyle s_{\bullet }=x_{h(\bullet )}}$ or ${\displaystyle s_{\bullet }=x_{\bullet }\circ h,}$ where ${\displaystyle \,\circ \,}$ denotes function composition and ${\displaystyle x_{h(\bullet )}:=\left(x_{h(i)}\right)_{i\in I}}$ is just notation for the function ${\displaystyle x_{\bullet }\circ h:I\to X.}$

Importantly, a subnet is not merely the restriction of a net ${\displaystyle \left(x_{a}\right)_{a\in A}}$ to a directed subset of its domain ${\displaystyle A.}$ In contrast, by definition, a subsequence of a given sequence ${\displaystyle x_{1},x_{2},x_{3},\ldots }$ is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. Explicitly, a sequence ${\displaystyle \left(s_{n}\right)_{n\in \mathbb {N} }}$ is said to be a subsequence of ${\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }}$ if there exists a strictly increasing sequence of positive integers ${\displaystyle h_{1}<h_{2}<h_{3}<\cdots }$ such that ${\displaystyle s_{n}=x_{h_{n}}}$ for every ${\displaystyle n\in \mathbb {N} }$ (that is to say, such that ${\displaystyle \left(s_{1},s_{2},\ldots \right)=\left(x_{h_{1}},x_{h_{2}},\ldots \right)}$). The sequence ${\displaystyle \left(h_{n}\right)_{n\in \mathbb {N} }=\left(h_{1},h_{2},\ldots \right)}$ can be canonically identified with the function ${\displaystyle h_{\bullet }:\mathbb {N} \to \mathbb {N} }$ defined by ${\displaystyle n\mapsto h_{n}.}$ Thus a sequence ${\displaystyle s_{\bullet }=\left(s_{n}\right)_{n\in \mathbb {N} }}$ is a subsequence of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in \mathbb {N} }}$ if and only if there exists a strictly increasing function ${\displaystyle h:\mathbb {N} \to \mathbb {N} }$ such that ${\displaystyle s_{\bullet }=x_{\bullet }\circ h.}$

Subsequences are subnets

Every subsequence is a subnet because if ${\displaystyle \left(x_{h_{n}}\right)_{n\in \mathbb {N} }}$ is a subsequence of ${\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }}$ then the map ${\displaystyle h:\mathbb {N} \to \mathbb {N} }$ defined by ${\displaystyle n\mapsto h_{n}}$ is an order-preserving map whose image is cofinal in its codomain and satisfies ${\displaystyle x_{h_{n}}=x_{h(n)}}$ for all ${\displaystyle n\in \mathbb {N} .}$