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 X = N {\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]
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 x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} and s ∙ = ( s i ) i ∈ I {\displaystyle s_{\bullet }=\left(s_{i}\right)_{i\in I}} are nets in a set X {\displaystyle X} from directed sets A {\displaystyle A} and I , {\displaystyle I,} respectively, then s ∙ {\displaystyle s_{\bullet }} is said to be a subnet of x ∙ {\displaystyle x_{\bullet }} (in the sense of Willard or a Willard–subnet[1]) if there exists a monotonefinal function h : I → A {\displaystyle h:I\to A} such that s i = x h ( i ) for all i ∈ I . {\displaystyle s_{i}=x_{h(i)}\quad {\text{ for all }}i\in I.} A function h : I → A {\displaystyle h:I\to A} is monotone, order-preserving, and an order homomorphism if whenever i ≤ j {\displaystyle i\leq j} then h ( i ) ≤ h ( j ) {\displaystyle h(i)\leq h(j)} and it is called final if its image h ( I ) {\displaystyle h(I)} is cofinal in A . {\displaystyle A.} The set h ( I ) {\displaystyle h(I)} being cofinal in A {\displaystyle A} means that for every a ∈ A , {\displaystyle a\in A,} there exists some b ∈ h ( I ) {\displaystyle b\in h(I)} such that b ≥ a ; {\displaystyle b\geq a;} that is, for every a ∈ A {\displaystyle a\in A} there exists an i ∈ I {\displaystyle i\in I} such that h ( i ) ≥ a . {\displaystyle h(i)\geq a.} [note 1]
Since the net x ∙ {\displaystyle x_{\bullet }} is the function x ∙ : A → X {\displaystyle x_{\bullet }:A\to X} and the net s ∙ {\displaystyle s_{\bullet }} is the function s ∙ : I → X , {\displaystyle s_{\bullet }:I\to X,} the defining condition ( s i ) i ∈ I = ( x h ( i ) ) i ∈ I , {\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 s ∙ = x h ( ∙ ) {\displaystyle s_{\bullet }=x_{h(\bullet )}} or s ∙ = x ∙ ∘ h , {\displaystyle s_{\bullet }=x_{\bullet }\circ h,} where ∘ {\displaystyle \,\circ \,} denotes function composition and x h ( ∙ ) := ( x h ( i ) ) i ∈ I {\displaystyle x_{h(\bullet )}:=\left(x_{h(i)}\right)_{i\in I}} is just notation for the function x ∙ ∘ h : I → X . {\displaystyle x_{\bullet }\circ h:I\to X.}
Importantly, a subnet is not merely the restriction of a net ( x a ) a ∈ A {\displaystyle \left(x_{a}\right)_{a\in A}} to a directed subset of its domain A . {\displaystyle A.} In contrast, by definition, a subsequence of a given sequence x 1 , x 2 , x 3 , … {\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 ( s n ) n ∈ N {\displaystyle \left(s_{n}\right)_{n\in \mathbb {N} }} is said to be a subsequence of ( x i ) i ∈ N {\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }} if there exists a strictly increasing sequence of positive integers h 1 < h 2 < h 3 < ⋯ {\displaystyle h_{1}<h_{2}<h_{3}<\cdots } such that s n = x h n {\displaystyle s_{n}=x_{h_{n}}} for every n ∈ N {\displaystyle n\in \mathbb {N} } (that is to say, such that ( s 1 , s 2 , … ) = ( x h 1 , x h 2 , … ) {\displaystyle \left(s_{1},s_{2},\ldots \right)=\left(x_{h_{1}},x_{h_{2}},\ldots \right)} ). The sequence ( h n ) n ∈ N = ( h 1 , h 2 , … ) {\displaystyle \left(h_{n}\right)_{n\in \mathbb {N} }=\left(h_{1},h_{2},\ldots \right)} can be canonically identified with the function h ∙ : N → N {\displaystyle h_{\bullet }:\mathbb {N} \to \mathbb {N} } defined by n ↦ h n . {\displaystyle n\mapsto h_{n}.} Thus a sequence s ∙ = ( s n ) n ∈ N {\displaystyle s_{\bullet }=\left(s_{n}\right)_{n\in \mathbb {N} }} is a subsequence of x ∙ = ( x i ) i ∈ N {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in \mathbb {N} }} if and only if there exists a strictly increasing function h : N → N {\displaystyle h:\mathbb {N} \to \mathbb {N} } such that s ∙ = x ∙ ∘ h . {\displaystyle s_{\bullet }=x_{\bullet }\circ h.}
Subsequences are subnets
Every subsequence is a subnet because if ( x h n ) n ∈ N {\displaystyle \left(x_{h_{n}}\right)_{n\in \mathbb {N} }} is a subsequence of ( x i ) i ∈ N {\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }} then the map h : N → N {\displaystyle h:\mathbb {N} \to \mathbb {N} } defined by n ↦ h n {\displaystyle n\mapsto h_{n}} is an order-preserving map whose image is cofinal in its codomain and satisfies x h n = x h ( n ) {\displaystyle x_{h_{n}}=x_{h(n)}} for all n ∈ N . {\displaystyle n\in \mathbb {N} .}
Sequence and subnet but not a subsequence
The sequence ( s i ) i ∈ N := ( 1 , 1 , 2 , 2 , 3 , 3 , … ) {\displaystyle \left(s_{i}\right)_{i\in \mathbb {N} }:=(1,1,2,2,3,3,\ldots )} is not a subsequence of ( x i ) i ∈ N := ( 1 , 2 , 3 , … ) {\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }:=(1,2,3,\ldots )} although it is a subnet because the map h : N → N {\displaystyle h:\mathbb {N} \to \mathbb {N} } defined by h ( i ) := ⌊ i + 1 2 ⌋ {\displaystyle h(i):=\left\lfloor {\tfrac {i+1}{2}}\right\rfloor } is an order-preserving map whose image is h ( N ) = N {\displaystyle h(\mathbb {N} )=\mathbb {N} } and satisfies s i = x h ( i ) {\displaystyle s_{i}=x_{h(i)}} for all i ∈ N . {\displaystyle i\in \mathbb {N} .} [note 2]
While a sequence is a net, a sequence has subnets that are not subsequences. 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.[2]
Subnet of a sequence that is not a sequence
A subnet of a sequence is not necessarily a sequence.[3]For an example, let I = { r ∈ R : r > 0 } {\displaystyle I=\{r\in \mathbb {R} :r>0\}} be directed by the usual order ≤ {\displaystyle \,\leq \,} and define h : I → N {\displaystyle h:I\to \mathbb {N} } by letting h ( r ) = ⌈ r ⌉ {\displaystyle h(r)=\lceil r\rceil } be the ceiling of r . {\displaystyle r.} Then h : ( I , ≤ ) → ( N , ≤ ) {\displaystyle h:(I,\leq )\to (\mathbb {N} ,\leq )} is an order-preserving map (because it is a non-decreasing function) whose image h ( I ) = N {\displaystyle h(I)=\mathbb {N} } is a cofinal subset of its codomain. Let x ∙ = ( x i ) i ∈ N : N → X {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in \mathbb {N} }:\mathbb {N} \to X} be any sequence (such as a constant sequence, for instance) and let s r := x h ( r ) {\displaystyle s_{r}:=x_{h(r)}} for every r ∈ I {\displaystyle r\in I} (in other words, let s ∙ := x ∙ ∘ h {\displaystyle s_{\bullet }:=x_{\bullet }\circ h} ). This net ( s r ) r ∈ I {\displaystyle \left(s_{r}\right)_{r\in I}} is not a sequence since its domain I {\displaystyle I} is an uncountable set. However, ( s r ) r ∈ I {\displaystyle \left(s_{r}\right)_{r\in I}} is a subnet of the sequence x ∙ {\displaystyle x_{\bullet }} since (by definition) s r = x h ( r ) {\displaystyle s_{r}=x_{h(r)}} holds for every r ∈ I . {\displaystyle r\in I.} Thus s ∙ {\displaystyle s_{\bullet }} is a subnet of x ∙ {\displaystyle x_{\bullet }} that is not a sequence.
Furthermore, the sequence x ∙ {\displaystyle x_{\bullet }} is also a subnet of ( s r ) r ∈ I {\displaystyle \left(s_{r}\right)_{r\in I}} since the inclusion map ι : N → I {\displaystyle \iota :\mathbb {N} \to I} (that sends n ↦ n {\displaystyle n\mapsto n} ) is an order-preserving map whose image ι ( N ) = N {\displaystyle \iota (\mathbb {N} )=\mathbb {N} } is a cofinal subset of its codomain and x n = s ι ( n ) {\displaystyle x_{n}=s_{\iota (n)}} holds for all n ∈ N . {\displaystyle n\in \mathbb {N} .} Thus x ∙ {\displaystyle x_{\bullet }} and ( s r ) r ∈ I {\displaystyle \left(s_{r}\right)_{r\in I}} are (simultaneously) subnets of each another.
Subnets induced by subsets
Suppose I ⊆ N {\displaystyle I\subseteq \mathbb {N} } is an infinite set and ( x i ) i ∈ N {\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }} is a sequence. Then ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} is a net on ( I , ≤ ) {\displaystyle (I,\leq )} that is also a subnet of ( x i ) i ∈ N {\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }} (take h : I → N {\displaystyle h:I\to \mathbb {N} } to be the inclusion map i ↦ i {\displaystyle i\mapsto i} ). This subnet ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} in turn induces a subsequence ( x h n ) n ∈ N {\displaystyle \left(x_{h_{n}}\right)_{n\in \mathbb {N} }} by defining h n {\displaystyle h_{n}} as the n th {\displaystyle n^{\text{th}}} smallest value in I {\displaystyle I} (that is, let h 1 := inf I {\displaystyle h_{1}:=\inf I} and let h n := inf { i ∈ I : i > h n − 1 } {\displaystyle h_{n}:=\inf\{i\in I:i>h_{n-1}\}} for every integer n > 1 {\displaystyle n>1} ). In this way, every infinite subset of I ⊆ N {\displaystyle I\subseteq \mathbb {N} } 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 h {\displaystyle h} be the identity map in the definition of "subnet" and requiring B {\displaystyle B} to be a cofinal subset of A {\displaystyle A} 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.
If s ∙ {\displaystyle s_{\bullet }} is a net in a subset S ⊆ X {\displaystyle S\subseteq X} and if x ∈ X {\displaystyle x\in X} is a cluster point of s ∙ {\displaystyle s_{\bullet }} then x ∈ cl X S . {\displaystyle x\in \operatorname {cl} _{X}S.} In other words, every cluster point of a net in a subset belongs to the closure of that set.
If x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} is a net in X {\displaystyle X} then the set of all cluster points of x ∙ {\displaystyle x_{\bullet }} in X {\displaystyle X} is equal to[3] ⋂ a ∈ A cl X ( x ≥ a ) {\displaystyle \bigcap _{a\in A}\operatorname {cl} _{X}\left(x_{\geq a}\right)} where x ≥ a := { x b : b ≥ a , b ∈ A } {\displaystyle x_{\geq a}:=\left\{x_{b}:b\geq a,b\in A\right\}} for each a ∈ A . {\displaystyle a\in A.}
If a net converges to a point x {\displaystyle x} then x {\displaystyle x} is necessarily a cluster point of that net.[3] The converse is not guaranteed in general. That is, it is possible for x ∈ X {\displaystyle x\in X} to be a cluster point of a net x ∙ {\displaystyle x_{\bullet }} but for x ∙ {\displaystyle x_{\bullet }} to not converge to x . {\displaystyle x.} However, if x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} clusters at x ∈ X {\displaystyle x\in X} then there exists a subnet of x ∙ {\displaystyle x_{\bullet }} that converges to x . {\displaystyle x.} This subnet can be explicitly constructed from ( A , ≤ ) {\displaystyle (A,\leq )} and the neighborhood filter N x {\displaystyle {\mathcal {N}}_{x}} at x {\displaystyle x} as follows: make I := { ( a , U ) ∈ A × N x : x a ∈ U } {\displaystyle I:=\left\{(a,U)\in A\times {\mathcal {N}}_{x}:x_{a}\in U\right\}} into a directed set by declaring that ( a , U ) ≤ ( b , V ) if and only if a ≤ b and U ⊇ V ; {\displaystyle (a,U)\leq (b,V)\quad {\text{ if and only if }}\quad a\leq b\;{\text{ and }}\;U\supseteq V;} then ( x a ) ( a , U ) ∈ I → x in X {\displaystyle \left(x_{a}\right)_{(a,U)\in I}\to x{\text{ in }}X} and ( x a ) ( a , U ) ∈ I {\displaystyle \left(x_{a}\right)_{(a,U)\in I}} is a subnet of x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} since the map α : I → A ( a , U ) ↦ a {\displaystyle {\begin{alignedat}{4}\alpha :\;&&I&&\;\to \;&A\\[0.3ex]&&(a,U)&&\;\mapsto \;&a\\\end{alignedat}}} is a monotone function whose image α ( I ) = A {\displaystyle \alpha (I)=A} is a cofinal subset of A , {\displaystyle A,} and x α ( ∙ ) := ( x α ( i ) ) i ∈ I = ( x α ( a , U ) ) ( a , U ) ∈ I = ( x a ) ( a , U ) ∈ I . {\displaystyle x_{\alpha (\bullet )}:=\left(x_{\alpha (i)}\right)_{i\in I}=\left(x_{\alpha (a,U)}\right)_{(a,U)\in I}=\left(x_{a}\right)_{(a,U)\in I}.}
Thus, a point x ∈ X {\displaystyle x\in X} is a cluster point of a given net if and only if it has a subnet that converges to x . {\displaystyle x.} [3]
Filter on a set – Family of subsets representing "large" sets
Filters in topology#Subnets – Use of filters to describe and characterize all basic topological notions and results
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