In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space. Nets are primarily used in the fields of analysis and topology, where they are used to characterize many important topological properties that (in general) sequences are unable to characterize (this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces). Nets are in one-to-one correspondence with filters.
History The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922. The term "net" was coined by John L. Kelley. The related concept of a filter was developed in 1937 by Henri Cartan.
Definitions A directed set is a non-empty set A {\displaystyle A} together with a preorder, typically automatically assumed to be denoted by ≤ {\displaystyle \,\leq \,} (unless indicated otherwise), with the property that it is also (upward) directed, which means that for any a , b ∈ A , {\displaystyle a,b\in A,} there exists some c ∈ A {\displaystyle c\in A} such that a ≤ c {\displaystyle a\leq c} and b ≤ c . {\displaystyle b\leq c.} In words, this property means that given any two elements (of A {\displaystyle A} ), there is always some element that is "above" both of them (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are not required to be total orders or even partial orders. A directed set may have a greatest element. In this case, the conditions a ≤ c {\displaystyle a\leq c} and b ≤ c {\displaystyle b\leq c} cannot be replaced by the strict inequalities a < c {\displaystyle a<c} and b < c {\displaystyle b<c} , since the strict inequalities cannot be satisfied if a or b is a greatest element. A net in X {\displaystyle X} , denoted x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} , is a function of the form x ∙ : A → X {\displaystyle x_{\bullet }:A\to X} whose domain A {\displaystyle A} is some directed set, and whose values are x ∙ ( a ) = x a {\displaystyle x_{\bullet }(a)=x_{a}} . Elements of a net's domain are called its indices. When the set X {\displaystyle X} is clear from context it is simply called a net, and one assumes A {\displaystyle A} is a directed set with preorder ≤ . {\displaystyle \,\leq .} Notation for nets varies, for example using angled brackets ⟨ x a ⟩ a ∈ A {\displaystyle \left\langle x_{a}\right\rangle _{a\in A}} . As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index a ∈ A {\displaystyle a\in A} .
Limits of nets
A net x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} is said to be eventually or residually in a set S {\displaystyle S} if there exists some a ∈ A {\displaystyle a\in A} such that for every b ∈ A {\displaystyle b\in A} with b ≥ a , {\displaystyle b\geq a,} the point x b ∈ S . {\displaystyle x_{b}\in S.} A point x ∈ X {\displaystyle x\in X} is called a limit point or limit of the net x ∙ {\displaystyle x_{\bullet }} in X {\displaystyle X} whenever:
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