Limits in a Cartesian product

A net in the product 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 their Cartesian product

$${\displaystyle {\textstyle \prod }X_{\bullet }:=\prod _{i\in I}X_{i}}$$

with the product topology, and that for every index ${\displaystyle l\in I,}$ denote the canonical projection to ${\displaystyle X_{l}}$ by

$${\displaystyle {\begin{alignedat}{4}\pi _{l}:\;&&{\textstyle \prod }X_{\bullet }&&\;\to \;&X_{l}\\[0.3ex]&&\left(x_{i}\right)_{i\in I}&&\;\mapsto \;&x_{l}\\\end{alignedat}}}$$

Let ${\displaystyle f_{\bullet }=\left(f_{a}\right)_{a\in A}}$ be a net in ${\displaystyle {\textstyle \prod }X_{\bullet }}$ directed by ${\displaystyle A}$ and for every index ${\displaystyle i\in I,}$ let

$${\displaystyle \pi _{i}\left(f_{\bullet }\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\pi _{i}\left(f_{a}\right)\right)_{a\in A}}$$

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 of function composition: the net ${\displaystyle \pi _{i}\left(f_{\bullet }\right)}$ is equal to the composition of the net ${\displaystyle f_{\bullet }:A\to {\textstyle \prod }X_{\bullet }}$ with the projection ${\displaystyle \pi _{i}:{\textstyle \prod }X_{\bullet }\to X_{i};}$ that is, ${\displaystyle \pi _{i}\left(f_{\bullet }\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\pi _{i}\,\circ \,f_{\bullet }.}$

For any given point ${\displaystyle L=\left(L_{i}\right)_{i\in I}\in {\textstyle \prod \limits _{i\in I}}X_{i},}$ the net ${\displaystyle f_{\bullet }}$ converges to ${\displaystyle L}$ in the product space ${\displaystyle {\textstyle \prod }X_{\bullet }}$ if and only if for every index ${\displaystyle i\in I,}$ ${\displaystyle \pi _{i}\left(f_{\bullet }\right)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\left(\pi _{i}\left(f_{a}\right)\right)_{a\in A}}$ converges to ${\displaystyle L_{i}}$ in ${\displaystyle X_{i}.}$^[6] And whenever the net ${\displaystyle f_{\bullet }}$ clusters at ${\displaystyle L}$ in ${\displaystyle {\textstyle \prod }X_{\bullet }}$ then ${\displaystyle \pi _{i}\left(f_{\bullet }\right)}$ clusters at ${\displaystyle L_{i}}$ for every index ${\displaystyle i\in I.}$^[7] However, the converse does not hold in general.^[7] 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 ${\displaystyle (1,1),(0,0),(1,1),(0,0),\ldots }$ 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 ${\displaystyle f_{\bullet }}$

Tychonoff's theorem and relation to the axiom of choice

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, the axiom of choice might be need to be assumed in order 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 the unique 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 a Hausdorff space. If ${\displaystyle I}$ is infinite and ${\displaystyle {\textstyle \prod }X_{\bullet }={\textstyle \prod \limits _{j\in I}}X_{j}}$ is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections ${\displaystyle \pi _{i}:{\textstyle \prod }X_{\bullet }\to X_{i}}$ are surjective maps.

The axiom of choice is equivalent to Tychonoff'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 the ultrafilter lemma and so strictly weaker than the axiom 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 convergent subnet.

Cluster points of a net

A point ${\displaystyle x\in X}$ is a cluster point of a given net if and only if it has a subset that converges to ${\displaystyle x.}$^[8] If ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ is a net in ${\displaystyle X}$ then the set of all cluster points of ${\displaystyle x_{\bullet }}$ in ${\displaystyle X}$ is equal to^[7]

$${\displaystyle \bigcap _{a\in A}\operatorname {cl} _{X}\left(x_{\geq a}\right)}$$

where ${\displaystyle x_{\geq a}:=\left\{x_{b}:b\geq a,b\in A\right\}}$ for each ${\displaystyle a\in A.}$ 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 ${\displaystyle x_{\bullet }.}$^[8]

Ultranets

A net ${\displaystyle x_{\bullet }}$ in set ${\displaystyle X}$ is called a universal net or an ultranet 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 ${\displaystyle X\setminus S.}$^[4] Ultranets are closely related to ultrafilters.

Every constant net is an ultranet. Every subnet of an ultranet is an ultranet.^[7] Every net has some subnet that is an ultranet.^[4] 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 ${\displaystyle Y.}$^[4]

Given ${\displaystyle x\in X,}$ an ultranet clusters at ${\displaystyle x}$ if and only it converges to ${\displaystyle x.}$^[4]

Examples of limits of nets

Every limit of a sequence and limit of a function can be interpreted as a limit of a net (as described below).

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

Interpret the set ${\displaystyle \mathbb {R} ^{\mathbb {R} }}$ of all functions with prototype ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ as the Cartesian product ${\displaystyle {\textstyle \prod \limits _{x\in \mathbb {R} }}\mathbb {R} }$ (by identifying a function ${\displaystyle f}$ with the tuple ${\displaystyle (f(x))_{x\in \mathbb {R} },}$ and conversely) and endow it with the product topology. This (product) topology on ${\displaystyle \mathbb {R} ^{\mathbb {R} }}$ is identical to the topology 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, ${\displaystyle \mathbf {0} \in \operatorname {cl} _{\mathbb {R} ^{\mathbb {R} }}E.}$^[7] This will be proven by constructing a net in ${\displaystyle E}$ that converges to ${\displaystyle \mathbf {0} .}$ However, there does not exist any sequence in ${\displaystyle E}$ that converges to ${\displaystyle \mathbf {0} ,}$^[9] 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 ${\displaystyle f\geq g}$ if and only if ${\displaystyle f(x)\geq g(x)}$ for all ${\displaystyle x.}$ This pointwise comparison is a partial order that makes ${\displaystyle (E,\geq )}$ 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 ${\displaystyle f\geq m}$ and ${\displaystyle g\geq m.}$ This partial order turns the identity map ${\displaystyle \operatorname {Id}$ :(E,\geq )\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 ${\displaystyle \mathbb {R} ^{\mathbb {R} }.}$

Sequence in a topological space

A sequence ${\displaystyle a_{1},a_{2},\ldots }$ in a topological space ${\displaystyle X}$ can be considered a net in ${\displaystyle X}$ defined on ${\displaystyle \mathbb {N} .}$

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 ${\displaystyle n\geq N,}$ 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 ${\displaystyle n\geq N}$ 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.

Function from a metric space to a topological space

Fix a point ${\displaystyle c\in M}$ in a metric space ${\displaystyle (M,d)}$ that has at least two point (such as ${\displaystyle M:=\mathbb {R} ^{n}}$ with the Euclidean 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 ${\displaystyle i\leq j}$ if and only if ${\displaystyle d(j,c)\leq d(i,c).}$ 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 ${\displaystyle (I,\leq ).}$^[7]

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 ${\displaystyle d(m,c)\leq d(n,c),}$ 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 ${\displaystyle \lim _{m\to c}f(m)\to L}$ in the usual sense (meaning that for every neighborhood ${\displaystyle V}$ of ${\displaystyle L,}$ ${\displaystyle f}$ is eventually in ${\displaystyle V}$).^[7]

The net ${\displaystyle f:M\setminus \{c\}\to X}$ is frequently in a subset ${\displaystyle S}$ of ${\displaystyle X}$ if and only if for every ${\displaystyle n\in M\setminus \{c\}}$ there exists some ${\displaystyle m\in M\setminus \{c\}}$ with ${\displaystyle d(m,c)\leq d(n,c)}$ such that ${\displaystyle f(m)}$ is in ${\displaystyle S.}$ Consequently, a point ${\displaystyle L\in X}$ is a cluster point of the net ${\displaystyle f}$ if and only if for every neighborhood ${\displaystyle V}$ of ${\displaystyle L,}$ the net is frequently in ${\displaystyle V.}$

Function from a well-ordered set to a topological space

Consider a well-ordered set ${\displaystyle [0,c]}$ with limit point ${\displaystyle t}$ and a function ${\displaystyle f}$ from ${\displaystyle [0,t)}$ to a topological space ${\displaystyle X.}$ This function is a net on ${\displaystyle [0,t).}$

It is eventually in a subset ${\displaystyle V}$ of ${\displaystyle X}$ if there exists an ${\displaystyle r\in [0,t)}$ such that for every ${\displaystyle s\in [r,t)}$ the point ${\displaystyle f(s)}$ is in ${\displaystyle V.}$

So ${\displaystyle \lim _{x\to t}f(x)\to L}$ if and only if for every neighborhood ${\displaystyle V}$ of ${\displaystyle L,}$ ${\displaystyle f}$ is eventually in ${\displaystyle V.}$

The net ${\displaystyle f}$ is frequently in a subset ${\displaystyle V}$ of ${\displaystyle X}$ if and only if for every ${\displaystyle r\in [0,t)}$ there exists some ${\displaystyle s\in [r,t)}$ such that ${\displaystyle f(s)\in V.}$

A point ${\displaystyle y\in X}$ is a cluster point of the net ${\displaystyle f}$ if and only if for every neighborhood ${\displaystyle V}$ of ${\displaystyle y,}$ the net is frequently in ${\displaystyle V.}$

The first example is a special case of this with ${\displaystyle c=\omega .}$

See also ordinal-indexed sequence.

Subnets

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,^[10] 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 a subnet or Willard-subnet^[10] of ${\displaystyle x_{\bullet }}$ if there exists an order-preserving map ${\displaystyle h:I\to A}$ such that ${\displaystyle h(I)}$ is a cofinal subset of ${\displaystyle A}$ and

$${\displaystyle s_{i}=x_{h(i)}\quad {\text{ for all }}i\in I.}$$

The map ${\displaystyle h:I\to A}$ is called order-preserving and an order homomorphism if whenever ${\displaystyle i\leq j}$ then ${\displaystyle h(i)\leq h(j).}$ 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 ${\displaystyle b\geq a.}$

Characterizations of topological properties

Closed sets and closure

A subset ${\displaystyle S\subseteq X}$ is closed in ${\displaystyle X}$ if and only if every limit point of every convergent net in ${\displaystyle S}$ necessarily belongs to ${\displaystyle S.}$ Explicitly, a subset ${\displaystyle S\subseteq X}$ is closed if and only if whenever ${\displaystyle x\in X}$ and ${\displaystyle s_{\bullet }=\left(s_{a}\right)_{a\in A}}$ is a net valued in ${\displaystyle S}$ (meaning that ${\displaystyle s_{a}\in S}$ for all ${\displaystyle a\in A}$) such that ${\displaystyle \lim {}_{}s_{\bullet }\to x}$ in ${\displaystyle X,}$ then necessarily ${\displaystyle x\in S.}$

More generally, if ${\displaystyle S\subseteq X}$ is any subset then a point ${\displaystyle x\in X}$ is in the closure of ${\displaystyle S}$ if and only if there exists a net ${\displaystyle \left(s_{a}\right)_{a\in A}}$ in ${\displaystyle S}$ with limit ${\displaystyle x\in X}$ and such that ${\displaystyle s_{a}\in S}$ for every index ${\displaystyle a\in A.}$^[8]

Open sets and characterizations of topologies

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 characterize topologies. 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.

Continuity

A function ${\displaystyle f:X\to Y}$ between topological spaces is continuous at a given point ${\displaystyle x}$ if and only if for every net ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ in its domain, if ${\displaystyle \lim _{}x_{\bullet }\to x}$ in ${\displaystyle X}$ then ${\displaystyle \lim {}f\left(x_{\bullet }\right)\to f(x)}$ in ${\displaystyle Y.}$^[8] Said more succinctly, a function ${\displaystyle f:X\to Y}$ is continuous if and only if whenever ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X}$ then ${\displaystyle f\left(x_{\bullet }\right)\to f(x)}$ 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 a first-countable space (or not a sequential space).

Compactness

A space ${\displaystyle X}$ is compact 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 the Bolzano–Weierstrass theorem and Heine–Borel theorem.

Proof

(${\displaystyle \implies }$) First, suppose that ${\displaystyle X}$ is compact. We will need the following observation (see finite intersection property). Let ${\displaystyle I}$ be any non-empty set and ${\displaystyle \left\{C_{i}\right\}_{i\in I}}$ be a collection of closed subsets of ${\displaystyle X}$ such that ${\displaystyle \bigcap _{i\in J}C_{i}\neq \varnothing }$ for each finite ${\displaystyle J\subseteq I.}$ Then ${\displaystyle \bigcap _{i\in I}C_{i}\neq \varnothing }$ as well. Otherwise, ${\displaystyle \left\{C_{i}^{c}\right\}_{i\in I}}$ would be an open cover for ${\displaystyle X}$ with no finite subcover contrary to the compactness of ${\displaystyle X.}$ Let ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ be a net in ${\displaystyle X}$ directed by ${\displaystyle A.}$ For every ${\displaystyle a\in A}$ define $${\displaystyle E_{a}\triangleq \left\{x_{b}:b\geq a\right\}.}$$ The collection ${\displaystyle \{\operatorname {cl} \left(E_{a}\right):a\in A\}}$ has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that $${\displaystyle \bigcap _{a\in A}\operatorname {cl} E_{a}\neq \varnothing }$$ and this is precisely the set of cluster points of ${\displaystyle x_{\bullet }.}$ By the proof given in the next section, it is equal to the set of limits of convergent subnets of ${\displaystyle x_{\bullet }.}$ Thus ${\displaystyle x_{\bullet }}$ has a convergent subnet. (${\displaystyle \Longleftarrow }$) Conversely, suppose that every net in ${\displaystyle X}$ has a convergent subnet. For the sake of contradiction, let ${\displaystyle \left\{U_{i}:i\in I\right\}}$ be an open cover of ${\displaystyle X}$ with no finite subcover. Consider ${\displaystyle D\triangleq \{J\subset I:|J|<\infty \}.}$ Observe that ${\displaystyle D}$ is a directed set under inclusion and for each ${\displaystyle C\in D,}$ there exists an ${\displaystyle x_{C}\in X}$ such that ${\displaystyle x_{C}\notin U_{a}}$ for all ${\displaystyle a\in C.}$ Consider the net ${\displaystyle \left(x_{C}\right)_{C\in D}.}$ This net cannot have a convergent subnet, because for each ${\displaystyle x\in X}$ there exists ${\displaystyle c\in I}$ such that ${\displaystyle U_{c}}$ is a neighbourhood of ${\displaystyle x}$; however, for all ${\displaystyle B\supseteq \{c\},}$ we have that ${\displaystyle x_{B}\notin U_{c}.}$ This is a contradiction and completes the proof.