Net (mathematics)

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 <math>A</math> together with a preorder, typically automatically assumed to be denoted by <math>\,\leq\,</math> (unless indicated otherwise), with the property that it is also () , which means that for any <math>a, b \in A,</math> there exists some <math>c \in A</math> such that <math>a \leq c</math> and <math>b \leq c.</math>

In words, this property means that given any two elements (of <math>A</math>), 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 required to be total orders or even partial orders. A directed set may have a greatest element. In this case, the conditions <math>a \leq c</math> and <math>b \leq c</math> cannot be replaced by the strict inequalities <math>a < c</math> and <math>b < c</math>, since the strict inequalities cannot be satisfied if a or b is a greatest element.

A net in <math>X</math>, denoted <math>x_\bull = \left(x_a\right)_{a \in A}</math>, is a function of the form <math>x_\bull : A \to X</math> whose domain <math>A</math> is some directed set, and whose values are <math>x_\bullet(a)= x_a </math>. Elements of a net's domain are called its . When the set <math>X</math> is clear from context it is simply called a net, and one assumes <math>A</math> is a directed set with preorder <math>\,\leq.</math> Notation for nets varies, for example using angled brackets <math>\left\langle x_a \right\rangle_{a \in A}</math>. As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index <math>a \in A</math>.

Limits of nets

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is said to be or a set <math>S</math> if there exists some <math>a \in A</math> such that for every <math>b \in A</math> with <math>b \geq a,</math> the point <math>x_b \in S.</math> A point <math>x \in X</math> is called a or of the net <math>x_\bull</math> in <math>X</math> whenever:

:for every open neighborhood <math>U</math> of <math>x,</math> the net <math>x_\bull</math> is eventually in <math>U</math>,

expressed equivalently as: the net or ; and variously denoted as:<math display="block">\begin{alignat}{4}

& x_\bull && \to\; && x && \;\;\text{ in } X \\

& x_a && \to\; && x && \;\;\text{ in } X \\

\lim \; & x_\bull && \to\; && x && \;\;\text{ in } X \\

\lim_{a \in A} \; & x_a && \to\; && x && \;\;\text{ in } X \\

\lim_a \; & x_a && \to\; && x && \;\;\text{ in } X.

\end{alignat}</math>If <math>X</math> is clear from context, it may be omitted from the notation.

If <math>\lim x_\bull \to x</math> and this limit is unique (i.e. <math>\lim x_\bull \to y</math> only for <math>x = y</math>) then one writes:<math display=block>\lim x_\bull = x \;~~ \text{ or } ~~\; \lim x_a = x \;~~ \text{ or } ~~\; \lim_{a \in A} x_a = x</math>using the equal sign in place of the arrow <math>\to.</math> In a Hausdorff space, the limit of a convergent net is unique.

Some authors do not distinguish between the notations <math>\lim x_\bull = x</math> and <math>\lim x_\bull \to x</math>, but this can lead to ambiguities if the ambient space <math>X</math> is not Hausdorff.

Cluster points of nets

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is said to be or <math>S</math> if for every <math>a \in A</math> there exists some <math>b \in A</math> such that <math>b \geq a</math> and <math>x_b \in S.</math> A point <math>x \in X</math> is said to be an or cluster point of a net if for every neighborhood <math>U</math> of <math>x,</math> the net is frequently/cofinally in <math>U.</math> In fact, <math>x \in X</math> is a cluster point if and only if it has a subnet that converges to <math>x.</math> The set <math display="inline">\operatorname{cl}_X \left( x_{\bullet} \right) </math> of all cluster points of <math>x_\bull</math> in <math>X</math> is equal to <math display="inline">\operatorname{cl}_X \left(x_{\geq a} \right) </math> for each <math>a\in A </math>, where <math>x_{\geq a} := \left\{x_b : b \geq a, b \in A\right\}</math>.

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, which is as follows:

If <math>x_\bull = \left(x_a\right)_{a \in A}</math> and <math>s_\bull = \left(s_i\right)_{i \in I}</math> are nets then <math>s_\bull</math> is called a or of <math>x_\bull</math> if there exists an order-preserving map <math>h : I \to A</math> such that <math>h(I)</math> is a cofinal subset of <math>A</math> and

The map <math>h : I \to A</math> is called and an if whenever <math>i \leq j</math> then <math>h(i) \leq h(j).</math>

The set <math>h(I)</math> being in <math>A</math> means that for every <math>a \in A,</math> there exists some <math>b \in h(I)</math> such that <math>b \geq a.</math>

If <math>x \in X</math> is a cluster point of some subnet of <math>x_\bull</math> then <math>x</math> is also a cluster point of <math>x_\bull.</math>

Ultranets

A net <math>x_\bull</math> in set <math>X</math> is called a or an if for every subset <math>S \subseteq X,</math> <math>x_\bull</math> is eventually in <math>S</math> or <math>x_\bull</math> is eventually in the complement <math>X \setminus S.</math>

Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet. Assuming the axiom of choice, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.

If <math>x_\bull = \left(x_a\right)_{a \in A}</math> is an ultranet in <math>X</math> and <math>f : X \to Y</math> is a function then <math>f \circ x_\bull = \left(f\left(x_a\right)\right)_{a \in A}</math> is an ultranet in <math>Y.</math>

Given <math>x \in X,</math> an ultranet clusters at <math>x</math> if and only if it converges to <math>x.</math>

Cauchy nets

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is a if for every entourage <math>V</math> there exists <math>c \in A</math> such that for all <math>a, b \geq c,</math> <math>\left(x_a, x_b\right)</math> is a member of <math>V.</math> More generally, in a Cauchy space, a net <math>x_\bull</math> is Cauchy if the filter generated by the net is a Cauchy filter.

A topological vector space (TVS) is called if every Cauchy net converges to some point. A normed space, which is a special type of topological vector space, is a complete TVS (equivalently, a Banach space) if and only if every Cauchy sequence converges to some point (a property that is called ). 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.

Characterizations of topological properties

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 of limit of a sequence. The following set of theorems and lemmas help cement that similarity:

Closed sets and closure

A subset <math>S \subseteq X</math> is closed in <math>X</math> if and only if every limit point in <math>X</math> of a net in <math>S</math> necessarily lies in <math>S</math>.

Explicitly, this means that if <math>s_\bull = \left(s_a\right)_{a \in A}</math> is a net with <math>s_a\in S </math> for all <math>a\in A </math>, and <math>\lim{}_{} s_\bull \to x</math> in <math>X,</math> then <math>x \in S.</math>

More generally, if <math>S \subseteq X</math> is any subset, the closure of <math>S</math> is the set of points <math>x\in X </math> with <math>\lim_{a\in A} s_\bullet \to x </math> for some net <math>\left(s_a\right)_{a \in A}</math> in <math>S</math>.

Open sets and characterizations of topologies

A subset <math>S \subseteq X</math> is open if and only if no net in <math>X \setminus S</math> converges to a point of <math>S.</math> Also, subset <math>S \subseteq X</math> is open if and only if every net converging to an element of <math>S</math> is eventually contained in <math>S.</math>

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 <math>f : X \to Y</math> between topological spaces is continuous at a point <math>x</math> if and only if for every net <math>x_\bull = \left(x_a\right)_{a \in A}</math> in the domain, <math>\lim_{} x_\bull \to x</math> in <math>X</math> implies <math>\lim{} f\left(x_\bull\right) \to f(x)</math> in <math>Y.</math>

Briefly, a function <math>f : X \to Y</math> is continuous if and only if <math>x_\bull \to x</math> in <math>X</math> implies <math>f\left(x_\bull\right) \to f(x)</math> in <math>Y.</math>

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 <math>X</math> is not a first-countable space (or not a sequential space).

(<math>\implies</math>)

Let <math>f</math> be continuous at point <math>x,</math> and let <math>x_\bull = \left(x_a\right)_{a \in A}</math> be a net such that <math>\lim_{} x_\bull \to x.</math>

Then for every open neighborhood <math>U</math> of <math>f(x),</math> its preimage under <math>f,</math> <math>V := f^{-1}(U),</math> is a neighborhood of <math>x</math> (by the continuity of <math>f</math> at <math>x</math>).

Thus the interior of <math>V,</math> which is denoted by <math>\operatorname{int} V,</math> is an open neighborhood of <math>x,</math> and consequently <math>x_\bull</math> is eventually in <math>\operatorname{int} V.</math> Therefore <math>\left(f\left(x_a\right)\right)_{a \in A}</math> is eventually in <math>f(\operatorname{int} V)</math> and thus also eventually in <math>f(V)</math> which is a subset of <math>U.</math> Thus <math>\lim_{} \left(f\left(x_a\right)\right)_{a \in A} \to f(x),</math> and this direction is proven.

(<math>\Longleftarrow</math>)

Let <math>x</math> be a point such that for every net <math>x_\bull = \left(x_a\right)_{a \in A}</math> such that <math>\lim_{} x_\bull \to x,</math> <math>\lim_{} \left(f\left(x_a\right)\right)_{a \in A} \to f(x).</math> Now suppose that <math>f</math> is not continuous at <math>x.</math>

Then there is a neighborhood <math>U</math> of <math>f(x)</math> whose preimage under <math>f,</math> <math>V,</math> is not a neighborhood of <math>x.</math> Because <math>f(x) \in U,</math> necessarily <math>x \in V.</math> Now the set of open neighborhoods of <math>x</math> with the containment preorder is a directed set (since the intersection of every two such neighborhoods is an open neighborhood of <math>x</math> as well).

We construct a net <math>x_\bull = \left(x_a\right)_{a \in A}</math> such that for every open neighborhood of <math>x</math> whose index is <math>a,</math> <math>x_a</math> is a point in this neighborhood that is not in <math>V</math>; that there is always such a point follows from the fact that no open neighborhood of <math>x</math> is included in <math>V</math> (because by assumption, <math>V</math> is not a neighborhood of <math>x</math>).

It follows that <math>f\left(x_a\right)</math> is not in <math>U.</math>

Now, for every open neighborhood <math>W</math> of <math>x,</math> this neighborhood is a member of the directed set whose index we denote <math>a_0.</math> For every <math>b \geq a_0,</math> the member of the directed set whose index is <math>b</math> is contained within <math>W</math>; therefore <math>x_b \in W.</math> Thus <math>\lim_{} x_\bull \to x.</math> and by our assumption <math>\lim_{} \left(f\left(x_a\right)\right)_{a \in A} \to f(x).</math>

But <math>\operatorname{int} U</math> is an open neighborhood of <math>f(x)</math> and thus <math>f\left(x_a\right)</math> is eventually in <math>\operatorname{int} U</math> and therefore also in <math>U,</math> in contradiction to <math>f\left(x_a\right)</math> not being in <math>U</math> for every <math>a.</math>

This is a contradiction so <math>f</math> must be continuous at <math>x.</math> This completes the proof.

Compactness

A space <math>X</math> is compact if and only if every net <math>x_\bull = \left(x_a\right)_{a \in A}</math> in <math>X</math> has a subnet with a limit in <math>X.</math> This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.

(<math>\implies</math>)

First, suppose that <math>X</math> is compact. We will need the following observation (see finite intersection property). Let <math>I</math> be any non-empty set and <math>\left\{C_i\right\}_{i \in I}</math> be a collection of closed subsets of <math>X</math> such that <math>\bigcap_{i \in J} C_i \neq \varnothing</math> for each finite <math>J \subseteq I.</math> Then <math>\bigcap_{i \in I} C_i \neq \varnothing</math> as well. Otherwise, <math>\left\{C_i^c\right\}_{i \in I}</math> would be an open cover for <math>X</math> with no finite subcover contrary to the compactness of <math>X.</math>

Let <math>x_\bull = \left(x_a\right)_{a \in A}</math> be a net in <math>X</math> directed by <math>A.</math> For every <math>a \in A</math> define

<math display=block>E_a \triangleq \left\{x_b : b \geq a\right\}.</math>

The collection <math>\{\operatorname{cl}\left(E_a\right) : a \in A\}</math> has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that

<math display=block>\bigcap_{a \in A} \operatorname{cl} E_a \neq \varnothing</math>

and this is precisely the set of cluster points of <math>x_\bull.</math> By the proof given in the next section, it is equal to the set of limits of convergent subnets of <math>x_\bull.</math> Thus <math>x_\bull</math> has a convergent subnet.

(<math>\Longleftarrow</math>)

Conversely, suppose that every net in <math>X</math> has a convergent subnet. For the sake of contradiction, let <math>\left\{U_i : i \in I\right\}</math> be an open cover of <math>X</math> with no finite subcover. Consider <math>D \triangleq \{J \subset I : |J| < \infty\}.</math> Observe that <math>D</math> is a directed set under inclusion and for each <math>C\in D,</math> there exists an <math>x_C \in X</math> such that <math>x_C \notin U_a</math> for all <math>a \in C.</math> Consider the net <math>\left(x_C\right)_{C \in D}.</math> This net cannot have a convergent subnet, because for each <math>x \in X</math> there exists <math>c \in I</math> such that <math>U_c</math> is a neighbourhood of <math>x</math>; however, for all <math>B \supseteq \{c\},</math> we have that <math>x_B \notin U_c.</math> This is a contradiction and completes the proof.

Cluster and limit points

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

Let <math>x_\bull = \left(x_a\right)_{a \in A}</math> be a net in a topological space <math>X</math> (where as usual <math>A</math> automatically assumed to be a directed set) and also let <math>y \in X.</math> If <math>y</math> is a limit of a subnet of <math>x_\bull</math> then <math>y</math> is a cluster point of <math>x_\bull.</math>

Conversely, assume that <math>y</math> is a cluster point of <math>x_\bull.</math>

Let <math>B</math> be the set of pairs <math>(U, a)</math> where <math>U</math> is an open neighborhood of <math>y</math> in <math>X</math> and <math>a \in A</math> is such that <math>x_a \in U.</math>

The map <math>h : B \to A</math> mapping <math>(U, a)</math> to <math>a</math> is then cofinal.

Moreover, giving <math>B</math> the product order (the neighborhoods of <math>y</math> are ordered by inclusion) makes it a directed set, and the net <math>\left(y_b\right)_{b \in B}</math> defined by <math>y_b = x_{h(b)}</math> converges to <math>y.</math>

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.

Other properties

In general, a net in a space <math>X</math> can have more than one limit, but if <math>X</math> is a Hausdorff space, the limit of a net, if it exists, is unique. Conversely, if <math>X</math> is not Hausdorff, then there exists a net on <math>X</math> with two distinct limits. Thus the uniqueness of the limit is 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 general preorder or partial order may have distinct limit points even in a Hausdorff space.

Relation to filters

A filter 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. More specifically, every filter base induces an using the filter's pointed sets, and convergence of the filter base implies convergence of the associated net. Similarly, any net <math>\left(x_a\right)_{a \in A}</math> in <math>X</math> induces a filter base of tails <math>\left\{\left\{x_a : a \in A, a_0 \leq a\right\} : a_0 \in A\right\}</math> where the filter in <math>X</math> generated by this filter base is called the net's . Convergence of the net implies convergence of the eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other. Some authors work even with more general structures than the real line, like complete lattices.

For a net <math>\left(x_a\right)_{a \in A},</math> put

<math display=block>\limsup x_a = \lim_{a \in A} \sup_{b \succeq a} x_b = \inf_{a \in A} \sup_{b \succeq a} x_b.</math>

Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,

<math display=block>\limsup (x_a + y_a) \leq \limsup x_a + \limsup y_a,</math>

where equality holds whenever one of the nets is convergent.

Riemann integral

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.

Metric spaces

Suppose <math>(M, d)</math> is a metric space (or a pseudometric space) and <math>M</math> is endowed with the metric topology. If <math>m \in M</math> is a point and <math>m_\bull = \left(m_i\right)_{a \in A}</math> is a net, then <math>m_\bull \to m</math> in <math>(M, d)</math> if and only if <math>d\left(m, m_\bull\right) \to 0</math> in <math>\R,</math> where <math>d\left(m, m_\bull\right) := \left(d\left(m, m_a\right)\right)_{a \in A}</math> is a net of real numbers.

In plain 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 <math>(M, \|\cdot\|)</math> is a normed space (or a seminormed space) then <math>m_\bull \to m</math> in <math>(M, \|\cdot\|)</math> if and only if <math>\left\|m - m_\bull\right\| \to 0</math> in <math>\Reals,</math> where <math>\left\|m - m_\bull\right\| := \left(\left\|m - m_a\right\|\right)_{a \in A}.</math>

If <math>(M, d)</math> has at least two points, then we can fix a point <math>c \in M</math> (such as <math>M := \R^n</math> with the Euclidean metric with <math>c := 0</math> being the origin, for example) and direct the set <math>I := M \setminus \{c\}</math> reversely according to distance from <math>c</math> by declaring that <math>i \leq j</math> if and only if <math>d(j, c) \leq d(i, c).</math> In other words, the relation is "has at least the same distance to <math>c</math> as", so that "large enough" with respect to this relation means "close enough to <math>c</math>".

Given any function with domain <math>M,</math> its restriction to <math>I := M \setminus \{c\}</math> can be canonically interpreted as a net directed by <math>(I, \leq).</math>

A net <math>f : M \setminus \{c\} \to X</math> is eventually in a subset <math>S</math> of a topological space <math>X</math> if and only if there exists some <math>n \in M \setminus \{c\}</math> such that for every <math>m \in M \setminus \{c\}</math> satisfying <math>d(m, c) \leq d(n, c),</math> the point <math>f(m)</math> is in <math>S.</math>

Such a net <math>f</math> converges in <math>X</math> to a given point <math>L \in X</math> if and only if <math>\lim_{m \to c} f(m) \to L</math> in the usual sense (meaning that for every neighborhood <math>V</math> of <math>L,</math> <math>f</math> is eventually in <math>V</math>).

The net <math>f : M \setminus \{c\} \to X</math> is frequently in a subset <math>S</math> of <math>X</math> if and only if for every <math>n \in M \setminus \{c\}</math> there exists some <math>m \in M \setminus \{c\}</math> with <math>d(m, c) \leq d(n, c)</math> such that <math>f(m)</math> is in <math>S.</math>

Consequently, a point <math>L \in X</math> is a cluster point of the net <math>f</math> if and only if for every neighborhood <math>V</math> of <math>L,</math> the net is frequently in <math>V.</math>

Function from a well-ordered set to a topological space

Consider a well-ordered set <math>[0, c]</math> with limit point <math>t</math> and a function <math>f</math> from <math>[0, t)</math> to a topological space <math>X.</math> This function is a net on <math>[0, t).</math>

It is eventually in a subset <math>V</math> of <math>X</math> if there exists an <math>r \in [0, t)</math> such that for every <math>s \in [r, t)</math> the point <math>f(s)</math> is in <math>V.</math>

So <math>\lim_{x \to t} f(x) \to L</math> if and only if for every neighborhood <math>V</math> of <math>L,</math> <math>f</math> is eventually in <math>V.</math>

The net <math>f</math> is frequently in a subset <math>V</math> of <math>X</math> if and only if for every <math>r \in [0, t)</math> there exists some <math>s \in [r, t)</math> such that <math>f(s) \in V.</math>

A point <math>y \in X</math> is a cluster point of the net <math>f</math> if and only if for every neighborhood <math>V</math> of <math>y,</math> the net is frequently in <math>V.</math>