In mathematics, more specifically ingeneral topology and related branches, anet orMoore–Smith sequence is afunction whose domain is adirected set. Thecodomain of this function is usually sometopological space. Nets directly generalize the concept of asequence in ametric space. Nets are primarily used in the fields ofanalysis andtopology, where they are used to characterize many importanttopological properties that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study ofsequential spaces andFréchet–Urysohn spaces). Nets are in one-to-one correspondence withfilters.

The concept of a net was first introduced byE. H. Moore andHerman L. Smith in 1922.^[1] The term "net" was coined byJohn L. Kelley.^[2][3]

The related concept of afilter was developed in 1937 byHenri Cartan.

Adirected set is a non-empty set${\displaystyle A}$  together with apreorder, typically automatically assumed to be denoted by${\displaystyle \le }$  (unless indicated otherwise), with the property that it is also (upward)directed, which means that for any${\displaystyle a,b\in A,}$  there exists some${\displaystyle c\in A}$  such that${\displaystyle a\le c}$  and${\displaystyle b\le c.}$  In words, this property means that given any two elements (of${\displaystyle A}$ ), 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 arenot required to betotal orders or evenpartial orders. A directed set may have thegreatest element. In this case, the conditions${\displaystyle a\le c}$  and${\displaystyle b\le c}$  cannot be replaced by the strict inequalities  and , since the strict inequalities cannot be satisfied ifa orb is the greatest element.

Anet in${\displaystyle X}$ , denoted${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$ , is afunction of the form${\displaystyle x_{\bullet }:A\to X}$  whosedomain${\displaystyle A}$  is some directed set, and whose values are${\displaystyle x_{\bullet }(a)=x_a}$ . Elements of a net's domain are called itsindices. When the set${\displaystyle X}$  is clear from context it is simply called anet, and one assumes${\displaystyle A}$  is a directed set with preorder${\displaystyle \le .}$  Notation for nets varies, for example using angled brackets${\displaystyle {⟨x_a⟩}_{a\in A}}$ . As is common inalgebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index${\displaystyle a\in A}$ .

A net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  is said to beeventually orresiduallyin a set${\displaystyle S}$  if there exists some${\displaystyle a\in A}$  such that for every${\displaystyle b\in A}$  with${\displaystyle b\ge a,}$  the point${\displaystyle x_b\in S.}$  A point${\displaystyle x\in X}$  is called alimit point orlimit of the net${\displaystyle x_{\bullet }}$  in${\displaystyle X}$  whenever:

for every openneighborhood${\displaystyle U}$  of${\displaystyle x,}$  the net${\displaystyle x_{\bullet }}$  is eventually in${\displaystyle U}$ ,

expressed equivalently as: the netconverges to/towards${\displaystyle x}$  orhas${\displaystyle x}$  as a limit; and variously denoted as: If${\displaystyle X}$  is clear from context, it may be omitted from the notation.

If${\displaystyle lim{x}_{\bullet }\to x}$  and this limit is unique (i.e.${\displaystyle lim{x}_{\bullet }\to y}$  only for${\displaystyle x=y}$ ) then one writes:$${\displaystyle lim{x}_{\bullet }=x\text{ or }lim{x}_a=x\text{ or }\underset{a\in A}{lim}{x}_a=x}$$ using the equal sign in place of the arrow${\displaystyle \to .}$ ^[4] In aHausdorff space, every net has at most one limit, and the limit of a convergent net is always unique.^[4]Some authors do not distinguish between the notations${\displaystyle lim{x}_{\bullet }=x}$  and${\displaystyle lim{x}_{\bullet }\to x}$ , but this can lead to ambiguities if the ambient space${\displaystyle X}$  is not Hausdorff.

A net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  is said to befrequently orcofinally in${\displaystyle S}$  if for every${\displaystyle a\in A}$  there exists some${\displaystyle b\in A}$  such that${\displaystyle b\ge a}$  and${\displaystyle x_b\in S.}$ ^[5] A point${\displaystyle x\in X}$  is said to be anaccumulation point orcluster point of a net if for every neighborhood${\displaystyle U}$  of${\displaystyle x,}$  the net is frequently/cofinally in${\displaystyle U.}$ ^[5] In fact,${\displaystyle x\in X}$  is a cluster point if and only if it has a subnet that converges to${\displaystyle x.}$ ^[6] The set${\mathrm{cl}}_X\left(x_{\bullet }\right)$  of all cluster points of${\displaystyle x_{\bullet }}$  in${\displaystyle X}$  is equal to${\mathrm{cl}}_X\left(x_{\ge a}\right)$  for each${\displaystyle a\in A}$ , where${\displaystyle x_{\ge a}:=\left\{x_b:b\ge a,b\in A\right\}}$ .

The analogue of "subsequence" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,^[7] 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 then${\displaystyle s_{\bullet }}$  is called asubnet orWillard-subnet^[7] of${\displaystyle x_{\bullet }}$  if there exists an order-preserving map${\displaystyle h:I\to A}$  such that${\displaystyle h(I)}$  is acofinal subset of${\displaystyle A}$  and$${\displaystyle s_i=x_{h(i)}\text{ for all }i\in I.}$$  The map${\displaystyle h:I\to A}$  is calledorder-preserving and anorder homomorphism if whenever${\displaystyle i\le j}$  then${\displaystyle h(i)\le h(j).}$  The set${\displaystyle h(I)}$  beingcofinal in${\displaystyle A}$  means that for every${\displaystyle a\in A,}$  there exists some${\displaystyle b\in h(I)}$  such that${\displaystyle b\ge a.}$ 

If${\displaystyle x\in X}$  is a cluster point of some subnet of${\displaystyle x_{\bullet }}$  then${\displaystyle x}$  is also a cluster point of${\displaystyle x_{\bullet }.}$ ^[6]

A net${\displaystyle x_{\bullet }}$  in set${\displaystyle X}$  is called auniversal net or anultranet if for every subset${\displaystyle S\subseteq X,}$ ${\displaystyle x_{\bullet }}$  is eventually in${\displaystyle S}$  or${\displaystyle x_{\bullet }}$  is eventually in the complement${\displaystyle X\setminus S.}$ ^[5]

Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet.^[8] Assuming theaxiom of choice, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.^[5] If${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  is an ultranet in${\displaystyle X}$  and${\displaystyle f:X\to Y}$  is a function then${\displaystyle f\circ x_{\bullet }={\left(f\left(x_a\right)\right)}_{a\in A}}$  is an ultranet in${\displaystyle Y.}$ ^[5]

Given${\displaystyle x\in X,}$  an ultranet clusters at${\displaystyle x}$  if and only it converges to${\displaystyle x.}$ ^[5]

A Cauchy net generalizes the notion ofCauchy sequence to nets defined onuniform spaces.^[9]

A net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  is aCauchy net if for everyentourage${\displaystyle V}$  there exists${\displaystyle c\in A}$  such that for all${\displaystyle a,b\ge c,}$ ${\displaystyle \left(x_a,x_b\right)}$  is a member of${\displaystyle V.}$ ^[9][10] More generally, in aCauchy space, a net${\displaystyle x_{\bullet }}$  is Cauchy if the filter generated by the net is aCauchy filter.

Atopological vector space (TVS) is calledcomplete if every Cauchy net converges to some point. Anormed space, which is a special type of topological vector space, is a complete TVS (equivalently, aBanach space) if and only if every Cauchy sequence converges to some point (a property that is calledsequential completeness). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-normable) topological vector spaces.

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that oflimit of a sequence. The following set of theorems and lemmas help cement that similarity:

A subset${\displaystyle S\subseteq X}$  is closed in${\displaystyle X}$  if and only if every limit point in${\displaystyle X}$  of a net in${\displaystyle S}$  necessarily lies in${\displaystyle S}$ .Explicitly, this means that if${\displaystyle s_{\bullet }={\left(s_a\right)}_{a\in A}}$  is a net with${\displaystyle s_a\in S}$  for all${\displaystyle a\in A}$ , and${\displaystyle lim{}_{}{s}_{\bullet }\to x}$  in${\displaystyle X,}$  then${\displaystyle x\in S.}$ 

More generally, if${\displaystyle S\subseteq X}$  is any subset, theclosure of${\displaystyle S}$  is the set of points${\displaystyle x\in X}$  with${\displaystyle \underset{a\in A}{lim}{s}_{\bullet }\to x}$  for some net${\displaystyle {\left(s_a\right)}_{a\in A}}$  in${\displaystyle S}$ .^[6]

A subset${\displaystyle S\subseteq X}$  is open if and only if no net in${\displaystyle X\setminus S}$  converges to a point of${\displaystyle S.}$ ^[11] Also, subset${\displaystyle S\subseteq X}$  is open if and only if every net converging to an element of${\displaystyle S}$  is eventually contained in${\displaystyle S.}$  It is these characterizations of "open subset" that allow nets to characterizetopologies. Topologies can also be characterized by closed subsets since a set is open if and only if its complement is closed. So the characterizations of "closed set" in terms of nets can also be used to characterize topologies.

A function${\displaystyle f:X\to Y}$  between topological spaces iscontinuous at a point${\displaystyle x}$  if and only if for every net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  in the domain,${\displaystyle \underset{}{lim}{x}_{\bullet }\to x}$  in${\displaystyle X}$  implies${\displaystyle limf\left(x_{\bullet }\right)\to f(x)}$  in${\displaystyle Y.}$ ^[6] Briefly, a function${\displaystyle f:X\to Y}$  is continuous if and only if${\displaystyle x_{\bullet }\to x}$  in${\displaystyle X}$  implies${\displaystyle f\left(x_{\bullet }\right)\to f(x)}$  in${\displaystyle Y.}$  In general, this statement would not be true if the word "net" was replaced by "sequence"; that is, it is necessary to allow for directed sets other than just the natural numbers if${\displaystyle X}$  is not afirst-countable space (or not asequential space).

A space${\displaystyle X}$  iscompact if and only if every net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$  in${\displaystyle X}$  has a subnet with a limit in${\displaystyle X.}$  This can be seen as a generalization of theBolzano–Weierstrass theorem andHeine–Borel theorem.

The set of cluster points of a net is equal to the set of limits of its convergentsubnets.

A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.

In general, a net in a space${\displaystyle X}$  can have more than one limit, but if${\displaystyle X}$  is aHausdorff space, the limit of a net, if it exists, is unique. Conversely, if${\displaystyle X}$  is not Hausdorff, then there exists a net on${\displaystyle X}$  with two distinct limits. Thus the uniqueness of the limit isequivalent to the Hausdorff condition on the space, and indeed this may be taken as the definition. This result depends on the directedness condition; a set indexed by a generalpreorder orpartial order may have distinct limit points even in a Hausdorff space.

Afilter is a related idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence.^[12] More specifically, everyfilter base induces anassociated net using the filter's pointed sets, and convergence of the filter base implies convergence of the associated net. Similarly, any net${\displaystyle {\left(x_a\right)}_{a\in A}}$  in${\displaystyle X}$  induces a filter base of tails${\displaystyle \left\{\left\{x_a:a\in A,a_0\le a\right\}:a_0\in A\right\}}$  where the filter in${\displaystyle X}$  generated by this filter base is called the net'seventuality filter. Convergence of the net implies convergence of the eventuality filter.^[13] This correspondence allows for any theorem that can be proven with one concept to be proven with the other.^[13] For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.

Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts.^[13] He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common inanalysis, while filters are most useful inalgebraic topology. In any case, he shows how the two can be used in combination to prove various theorems ingeneral topology.

The learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especiallyanalysts, prefer them over filters. However, filters, and especiallyultrafilters, have some important technical advantages over nets that ultimately result in nets being encountered much less often than filters outside of the fields of analysis and topology.

Every non-emptytotally ordered set is directed. Therefore, every function on such a set is a net. In particular, thenatural numbers${\displaystyle \mathbb{N}}$  together with the usual integer comparison${\displaystyle \le }$  preorder form thearchetypical example of a directed set. A sequence is a function on the natural numbers, so every sequence${\displaystyle a_1,a_2,\dots }$  in a topological space${\displaystyle X}$  can be considered a net in${\displaystyle X}$  defined on${\displaystyle \mathbb{N}.}$  Conversely, any net whose domain is the natural numbers is asequence because by definition, a sequence in${\displaystyle X}$  is just a function from${\displaystyle \mathbb{N}=\{1,2,\dots \}}$  into${\displaystyle X.}$  It is in this way that nets are generalizations of sequences: rather than being defined on acountablelinearly ordered set (${\displaystyle \mathbb{N}}$ ), a net is defined on an arbitrarydirected set. Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences. For example, the subscript notation${\displaystyle x_a}$  is taken from sequences.

Similarly, everylimit of a sequence andlimit of a function can be interpreted as a limit of a net. Specifically, the net is eventually in a subset${\displaystyle S}$  of${\displaystyle X}$  if there exists an${\displaystyle N\in \mathbb{N}}$  such that for every integer${\displaystyle n\ge N,}$  the point${\displaystyle a_n}$  is in${\displaystyle S.}$  So${\displaystyle lim{}_n{a}_n\to L}$  if and only if for every neighborhood${\displaystyle V}$  of${\displaystyle L,}$  the net is eventually in${\displaystyle V.}$  The net is frequently in a subset${\displaystyle S}$  of${\displaystyle X}$  if and only if for every${\displaystyle N\in \mathbb{N}}$  there exists some integer${\displaystyle n\ge N}$  such that${\displaystyle a_n\in S,}$  that is, if and only if infinitely many elements of the sequence are in${\displaystyle S.}$  Thus a point${\displaystyle y\in X}$  is a cluster point of the net if and only if every neighborhood${\displaystyle V}$  of${\displaystyle y}$  contains infinitely many elements of the sequence.

In the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map${\displaystyle f}$  between topological spaces${\displaystyle X}$  and${\displaystyle Y}$ :

1. The map${\displaystyle f}$  iscontinuous in the topological sense;
2. Given any point${\displaystyle x}$  in${\displaystyle X,}$  and any sequence in${\displaystyle X}$  converging to${\displaystyle x,}$  the composition of${\displaystyle f}$  with this sequence converges to${\displaystyle f(x)}$ (continuous in the sequential sense).

While condition 1 always guarantees condition 2, the converse is not necessarily true. The spaces for which the two conditions are equivalent are calledsequential spaces. Allfirst-countable spaces, includingmetric spaces, are sequential spaces, but not all topological spaces are sequential. Nets generalize the notion of a sequence so that condition 2 reads as follows:

2. Given any point${\displaystyle x}$  in${\displaystyle X,}$  and any net in${\displaystyle X}$  converging to${\displaystyle x,}$  the composition of${\displaystyle f}$  with this net converges to${\displaystyle f(x)}$  (continuous in the net sense).

With this change, the conditions become equivalent for all maps of topological spaces, including topological spaces that do not necessarily have a countable or linearly orderedneighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much likedirected sets in behavior.

For an example where sequences do not suffice, interpret the set${\displaystyle {\mathbb{R}}^{\mathbb{R}}}$  of all functions with prototype${\displaystyle f:\mathbb{R}\to \mathbb{R}}$  as the Cartesian product${\displaystyle \prod _{x\in \mathbb{R}}\mathbb{R}}$  (by identifying a function${\displaystyle f}$  with the tuple${\displaystyle (f(x){)}_{x\in \mathbb{R}},}$  and conversely) and endow it with theproduct topology. This (product) topology on${\displaystyle {\mathbb{R}}^{\mathbb{R}}}$  is identical to thetopology of pointwise convergence. Let${\displaystyle E}$  denote the set of all functions${\displaystyle f:\mathbb{R}\to \{0,1\}}$  that are equal to${\displaystyle 1}$  everywhere except for at most finitely many points (that is, such that the set${\displaystyle \{x:f(x)=0\}}$  is finite). Then the constant${\displaystyle 0}$  function${\displaystyle \mathbf{0}:\mathbb{R}\to \{0\}}$  belongs to the closure of${\displaystyle E}$  in${\displaystyle {\mathbb{R}}^{\mathbb{R}};}$  that is,${\displaystyle \mathbf{0}\in {\mathrm{cl}}_{{\mathbb{R}}^{\mathbb{R}}}E.}$ ^[8] This will be proven by constructing a net in${\displaystyle E}$  that converges to${\displaystyle \mathbf{0}.}$  However, there does not exist anysequence in${\displaystyle E}$  that converges to${\displaystyle \mathbf{0},}$ ^[14] which makes this one instance where (non-sequence) nets must be used because sequences alone can not reach the desired conclusion. Compare elements of${\displaystyle {\mathbb{R}}^{\mathbb{R}}}$  pointwise in the usual way by declaring that${\displaystyle f\ge g}$  if and only if${\displaystyle f(x)\ge g(x)}$  for all${\displaystyle x.}$  This pointwise comparison is a partial order that makes${\displaystyle (E,\ge )}$  a directed set since given any${\displaystyle f,g\in E,}$  their pointwise minimum${\displaystyle m:=min\{f,g\}}$  belongs to${\displaystyle E}$  and satisfies${\displaystyle f\ge m}$  and${\displaystyle g\ge m.}$  This partial order turns theidentity map${\displaystyle \mathrm{Id}:(E,\ge )\to E}$  (defined by${\displaystyle f\mapsto f}$ ) into an${\displaystyle E}$ -valued net. This net converges pointwise to${\displaystyle \mathbf{0}}$  in${\displaystyle {\mathbb{R}}^{\mathbb{R}},}$  which implies that${\displaystyle \mathbf{0}}$  belongs to the closure of${\displaystyle E}$  in${\displaystyle {\mathbb{R}}^{\mathbb{R}}.}$ 

More generally, a subnet of a sequence isnot necessarily a sequence.^[5][a] Moreso, a subnet of a sequence may be a sequence, but not a subsequence.^[b] But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences. Specifically, for a first-countable space, the net${\displaystyle {\left(x_a\right)}_{a\in A}}$  induces the sequence${\displaystyle {\left(x_{h_n}\right)}_{n\in \mathbb{N}}}$  where${\displaystyle h_n}$  is defined as the${\displaystyle n^{\text{th}}}$  smallest value in${\displaystyle A}$  – that is, let${\displaystyle h_1:=infA}$  and let${\displaystyle h_n:=inf\{a\in A:a>h_{n-1}\}}$  for every integer${\displaystyle n>1}$ .

If the set${\displaystyle S=\{x\}\cup \left\{x_a:a\in A\right\}}$  is endowed with thesubspace topology induced on it by${\displaystyle X,}$  then${\displaystyle \underset{}{lim}{x}_{\bullet }\to x}$  in${\displaystyle X}$  if and only if${\displaystyle \underset{}{lim}{x}_{\bullet }\to x}$  in${\displaystyle S.}$  In this way, the question of whether or not the net${\displaystyle x_{\bullet }}$  converges to the given point${\displaystyle x}$  dependssolely on this topological subspace${\displaystyle S}$  consisting of${\displaystyle x}$  and theimage of (that is, the points of) the net${\displaystyle x_{\bullet }.}$ 

Intuitively, convergence of a net${\displaystyle {\left(x_a\right)}_{a\in A}}$  means that the values${\displaystyle x_a}$  come and stay as close as we want to${\displaystyle x}$  for large enough${\displaystyle a.}$  Given a point${\displaystyle x}$  in a topological space, let${\displaystyle N_x}$  denote the set of allneighbourhoods containing${\displaystyle x.}$  Then${\displaystyle N_x}$  is a directed set, where the direction is given by reverse inclusion, so that${\displaystyle S\ge T}$ if and only if${\displaystyle S}$  is contained in${\displaystyle T.}$  For${\displaystyle S\in N_x,}$  let${\displaystyle x_S}$  be a point in${\displaystyle S.}$  Then${\displaystyle \left(x_S\right)}$  is a net. As${\displaystyle S}$  increases with respect to${\displaystyle \ge ,}$  the points${\displaystyle x_S}$  in the net are constrained to lie in decreasing neighbourhoods of${\displaystyle x,}$ . Therefore, in thisneighborhood system of a point${\displaystyle x}$ ,${\displaystyle x_S}$  does indeed converge to${\displaystyle x}$  according to the definition of net convergence.

Given asubbase${\displaystyle \mathcal{B}}$  for the topology on${\displaystyle X}$  (where note that everybase for a topology is also a subbase) and given a point${\displaystyle x\in X,}$  a net${\displaystyle x_{\bullet }}$  in${\displaystyle X}$  converges to${\displaystyle x}$  if and only if it is eventually in every neighborhood${\displaystyle U\in \mathcal{B}}$  of${\displaystyle x.}$  This characterization extends toneighborhood subbases (and so alsoneighborhood bases) of the given point${\displaystyle x.}$ 

A net in theproduct space has a limit if and only if each projection has a limit.

Explicitly, let${\displaystyle {\left(X_i\right)}_{i\in I}}$  be topological spaces, endow theirCartesian product$${\displaystyle \prod X_{\bullet }:=\prod _{i\in I}{X}_i}$$ with theproduct topology, and that for every index${\displaystyle l\in I,}$  denote the canonical projection to${\displaystyle X_l}$  by 

Let${\displaystyle f_{\bullet }={\left(f_a\right)}_{a\in A}}$  be a net in${\displaystyle \prod X_{\bullet }}$  directed by${\displaystyle A}$  and for every index${\displaystyle i\in I,}$  let$${\displaystyle {\pi }_i\left(f_{\bullet }\right)\stackrel{\text{def}}{=}{\left({\pi }_i\left(f_a\right)\right)}_{a\in A}}$$ denote the result of "plugging${\displaystyle f_{\bullet }}$  into${\displaystyle {\pi }_i}$ ", which results in the net${\displaystyle {\pi }_i\left(f_{\bullet }\right):A\to X_i.}$  It is sometimes useful to think of this definition in terms offunction composition: the net${\displaystyle {\pi }_i\left(f_{\bullet }\right)}$  is equal to the composition of the net${\displaystyle f_{\bullet }:A\to \prod X_{\bullet }}$  with the projection${\displaystyle {\pi }_i:\prod X_{\bullet }\to X_i;}$  that is,${\displaystyle {\pi }_i\left(f_{\bullet }\right)\stackrel{\text{def}}{=}{\pi }_i\circ f_{\bullet }.}$ 

For any given point${\displaystyle L={\left(L_i\right)}_{i\in I}\in \prod _{i\in I}{X}_i,}$  the net${\displaystyle f_{\bullet }}$  converges to${\displaystyle L}$  in the product space${\displaystyle \prod X_{\bullet }}$  if and only if for every index${\displaystyle i\in I,}$ ${\displaystyle {\pi }_i\left(f_{\bullet }\right)\stackrel{\text{def}}{=}{\left({\pi }_i\left(f_a\right)\right)}_{a\in A}}$  converges to${\displaystyle L_i}$  in${\displaystyle X_i.}$ ^[15] And whenever the net${\displaystyle f_{\bullet }}$  clusters at${\displaystyle L}$  in${\displaystyle \prod X_{\bullet }}$  then${\displaystyle {\pi }_i\left(f_{\bullet }\right)}$  clusters at${\displaystyle L_i}$  for every index${\displaystyle i\in I.}$ ^[8] However, the converse does not hold in general.^[8] For example, suppose${\displaystyle X_1=X_2=\mathbb{R}}$  and let${\displaystyle f_{\bullet }={\left(f_a\right)}_{a\in \mathbb{N}}}$  denote the sequence${\displaystyle (1,1),(0,0),(1,1),(0,0),\dots }$  that alternates between${\displaystyle (1,1)}$  and${\displaystyle (0,0).}$  Then${\displaystyle L_1:=0}$  and${\displaystyle L_2:=1}$  are cluster points of both${\displaystyle {\pi }_1\left(f_{\bullet }\right)}$  and${\displaystyle {\pi }_2\left(f_{\bullet }\right)}$  in${\displaystyle X_1\times X_2={\mathbb{R}}^2}$  but${\displaystyle \left(L_1,L_2\right)=(0,1)}$  is not a cluster point of${\displaystyle f_{\bullet }}$  since the open ball of radius${\displaystyle 1}$  centered at${\displaystyle (0,1)}$  does not contain even a single point${\displaystyle f_{\bullet }}$ 

If no${\displaystyle L\in X}$  is given but for every${\displaystyle i\in I,}$  there exists some${\displaystyle L_i\in X_i}$  such that${\displaystyle {\pi }_i\left(f_{\bullet }\right)\to L_i}$  in${\displaystyle X_i}$  then the tuple defined by${\displaystyle L={\left(L_i\right)}_{i\in I}}$  will be a limit of${\displaystyle f_{\bullet }}$  in${\displaystyle X.}$  However, theaxiom of choice might be need to be assumed to conclude that this tuple${\displaystyle L}$  exists; the axiom of choice is not needed in some situations, such as when${\displaystyle I}$  is finite or when every${\displaystyle L_i\in X_i}$  is theunique limit of the net${\displaystyle {\pi }_i\left(f_{\bullet }\right)}$  (because then there is nothing to choose between), which happens for example, when every${\displaystyle X_i}$  is aHausdorff space. If${\displaystyle I}$  is infinite and${\displaystyle \prod X_{\bullet }=\prod _{j\in I}{X}_j}$  is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections${\displaystyle {\pi }_i:\prod X_{\bullet }\to X_i}$  aresurjective maps.

The axiom of choice is equivalent toTychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. But if every compact space is also Hausdorff, then the so called "Tychonoff's theorem for compact Hausdorff spaces" can be used instead, which is equivalent to theultrafilter lemma and so strictly weaker than theaxiom of choice. Nets can be used to give short proofs of both version of Tychonoff's theorem by using the characterization of net convergence given above together with the fact that a space is compact if and only if every net has a convergentsubnet.

Limit superior andlimit inferior of a net of real numbers can be defined in a similar manner as for sequences.^[16][17][18] Some authors work even with more general structures than the real line, like complete lattices.^[19]

For a net${\displaystyle {\left(x_a\right)}_{a\in A},}$  put$${\displaystyle lim sup{x}_a=\underset{a\in A}{lim}\underset{b⪰a}{sup}{x}_b=\underset{a\in A}{inf}\underset{b⪰a}{sup}{x}_b.}$$ 

Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,$${\displaystyle lim sup(x_a+y_a)\le lim sup{x}_a+lim sup{y}_a,}$$ where equality holds whenever one of the nets is convergent.

The definition of the value of aRiemann integral can be interpreted as a limit of a net ofRiemann sums where the net's directed set is the set of allpartitions of the interval of integration, partially ordered by inclusion.

Suppose${\displaystyle (M,d)}$  is ametric space (or apseudometric space) and${\displaystyle M}$  is endowed with themetric topology. If${\displaystyle m\in M}$  is a point and${\displaystyle m_{\bullet }={\left(m_i\right)}_{a\in A}}$  is a net, then${\displaystyle m_{\bullet }\to m}$  in${\displaystyle (M,d)}$  if and only if${\displaystyle d\left(m,m_{\bullet }\right)\to 0}$  in${\displaystyle \mathbb{R},}$  where${\displaystyle d\left(m,m_{\bullet }\right):={\left(d\left(m,m_a\right)\right)}_{a\in A}}$  is a net ofreal numbers. Inplain English, this characterization says that a net converges to a point in a metric space if and only if the distance between the net and the point converges to zero. If${\displaystyle (M,\Vert \cdot \Vert )}$  is anormed space (or aseminormed space) then${\displaystyle m_{\bullet }\to m}$  in${\displaystyle (M,\Vert \cdot \Vert )}$  if and only if${\displaystyle \Vert m-m_{\bullet }\Vert \to 0}$  in${\displaystyle \mathbb{R},}$  where${\displaystyle \Vert m-m_{\bullet }\Vert :={\left(\Vert m-m_a\Vert \right)}_{a\in A}.}$ 

If${\displaystyle (M,d)}$  has at least two points, then we can fix a point${\displaystyle c\in M}$  (such as${\displaystyle M:={\mathbb{R}}^n}$  with theEuclidean metric with${\displaystyle c:=0}$  being the origin, for example) and direct the set${\displaystyle I:=M\setminus \{c\}}$  reversely according to distance from${\displaystyle c}$  by declaring that${\displaystyle i\le j}$  if and only if${\displaystyle d(j,c)\le d(i,c).}$  In other words, the relation is "has at least the same distance to${\displaystyle c}$  as", so that "large enough" with respect to this relation means "close enough to${\displaystyle c}$ ". Given any function with domain${\displaystyle M,}$  its restriction to${\displaystyle I:=M\setminus \{c\}}$  can be canonically interpreted as a net directed by${\displaystyle (I,\le ).}$ ^[8]

A net${\displaystyle f:M\setminus \{c\}\to X}$  is eventually in a subset${\displaystyle S}$  of a topological space${\displaystyle X}$  if and only if there exists some${\displaystyle n\in M\setminus \{c\}}$  such that for every${\displaystyle m\in M\setminus \{c\}}$  satisfying${\displaystyle d(m,c)\le d(n,c),}$  the point${\displaystyle f(m)}$  is in${\displaystyle S.}$  Such a net${\displaystyle f}$  converges in${\displaystyle X}$  to a given point${\displaystyle L\in X}$  if and only if${\displaystyle \underset{m\to c}{lim}f(m)\to L}$  in the usual sense (meaning that for every neighborhood${\displaystyle V}$  of${\displaystyle L,}$ ${\displaystyle f}$  is eventually in${\displaystyle V}$ ).^[8]