Editing Net (mathematics) (section)

==Examples==

=== Subspace topology ===

If the set <math>S = \{x\} \cup \left\{x_a : a \in A\right\}</math> is endowed with the [[subspace topology]] induced on it by <math>X,</math> then <math>\lim_{} x_\bull \to x</math> in <math>X</math> if and only if <math>\lim_{} x_\bull \to x</math> in <math>S.</math> In this way, the question of whether or not the net <math>x_\bull</math> converges to the given point <math>x</math> depends {{em|solely}} on this topological subspace <math>S</math> consisting of <math>x</math> and the [[Image (mathematics)|image]] of (that is, the points of) the net <math>x_\bull.</math>

=== Neighborhood systems ===
{{Main|Neighborhood system}}

Intuitively, convergence of a net <math>\left(x_a\right)_{a \in A}</math> means that the values <math>x_a</math> come and stay as close as we want to <math>x</math> for large enough <math>a.</math> Given a point <math>x</math> in a topological space, let <math>N_x</math> denote the set of all [[Neighbourhood (topology)|neighbourhood]]s containing <math>x.</math> Then <math>N_x</math> is a directed set, where the direction is given by reverse inclusion, so that <math>S \geq T</math> [[if and only if]] <math>S</math> is contained in <math>T.</math> For <math>S \in N_x,</math> let <math>x_S</math> be a point in <math>S.</math> Then <math>\left(x_S\right)</math> is a net. As <math>S</math> increases with respect to <math>\,\geq,</math> the points <math>x_S</math> in the net are constrained to lie in decreasing neighbourhoods of <math>x,</math>. Therefore, in this [[neighborhood system]] of a point <math>x</math>, <math>x_S</math> does indeed converge to <math>x</math> according to the definition of net convergence.

Given a [[subbase]] <math>\mathcal{B}</math> for the topology on <math>X</math> (where note that every [[Base (topology)|base]] for a topology is also a subbase) and given a point <math>x \in X,</math> a net <math>x_\bull</math> in <math>X</math> converges to <math>x</math> if and only if it is eventually in every neighborhood <math>U \in \mathcal{B}</math> of <math>x.</math> This characterization extends to [[Neighbourhood system|neighborhood subbases]] (and so also [[Neighbourhood system|neighborhood bases]]) of the given point <math>x.</math>

===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 <math>\left(X_i\right)_{i \in I}</math> be topological spaces, endow their [[Cartesian product]] 
<math display=block>{\textstyle\prod} X_\bull := \prod_{i \in I} X_i</math>
with the [[product topology]], and that for every index <math>l \in I,</math> denote the canonical projection to <math>X_l</math> by
<math display=block>\begin{alignat}{4}
\pi_l :\;&& {\textstyle\prod} X_\bull  &&\;\to\;& X_l \\[0.3ex]
         && \left(x_i\right)_{i \in I} &&\;\mapsto\;& x_l \\
\end{alignat}</math>

Let <math>f_\bull = \left(f_a\right)_{a \in A}</math> be a net in <math>{\textstyle\prod} X_\bull</math> directed by <math>A</math> and for every index <math>i \in I,</math> let 
<math display=block>\pi_i\left(f_\bull\right) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\pi_i\left(f_a\right)\right)_{a \in A}</math>
denote the result of "plugging <math>f_\bull</math> into <math>\pi_i</math>", which results in the net <math>\pi_i\left(f_\bull\right) : A \to X_i.</math> 
It is sometimes useful to think of this definition in terms of [[function composition]]: the net <math>\pi_i\left(f_\bull\right)</math> is equal to the composition of the net <math>f_\bull : A \to {\textstyle\prod} X_\bull</math> with the projection <math>\pi_i : {\textstyle\prod} X_\bull \to X_i;</math> that is, <math>\pi_i\left(f_\bull\right) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \pi_i \,\circ\, f_\bull.</math>

For any given point <math>L = \left(L_i\right)_{i \in I} \in {\textstyle\prod\limits_{i \in I}} X_i,</math> the net <math>f_\bull</math> converges to <math>L</math> in the product space <math>{\textstyle\prod} X_\bull</math> if and only if for every index <math>i \in I,</math> <math>\pi_i\left(f_\bull\right) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \left(\pi_i\left(f_a\right)\right)_{a \in A}</math> converges to <math>L_i</math> in <math>X_i.</math>{{sfn|Willard|2004|p=76}} 
And whenever the net <math>f_\bull</math> clusters at <math>L</math> in <math>{\textstyle\prod} X_\bull</math> then <math>\pi_i\left(f_\bull\right)</math> clusters at <math>L_i</math> for every index <math>i \in I.</math>{{sfn|Willard|2004|p=77}} However, the converse does not hold in general.{{sfn|Willard|2004|p=77}} For example, suppose <math>X_1 = X_2 = \Reals</math> and let <math>f_\bull = \left(f_a\right)_{a \in \N}</math> denote the sequence <math>(1, 1), (0, 0), (1, 1), (0, 0), \ldots</math> that alternates between <math>(1, 1)</math> and <math>(0, 0).</math> Then <math>L_1 := 0</math> and <math>L_2 := 1</math> are cluster points of both <math>\pi_1\left(f_\bull\right)</math> and <math>\pi_2\left(f_\bull\right)</math> in <math>X_1 \times X_2 = \Reals^2</math> but <math>\left(L_1, L_2\right) = (0, 1)</math> is not a cluster point of <math>f_\bull</math> since the open ball of radius <math>1</math> centered at <math>(0, 1)</math> does not contain even a single point <math>f_\bull</math> 

=== Tychonoff's theorem and relation to the axiom of choice ===

If no <math>L \in X</math> is given but for every <math>i \in I,</math> there exists some <math>L_i \in X_i</math> such that <math>\pi_i\left(f_\bull\right) \to L_i</math> in <math>X_i</math> then the tuple defined by <math>L = \left(L_i\right)_{i \in I}</math> will be a limit of <math>f_\bull</math> in <math>X.</math> 
However, the [[axiom of choice]] might be need to be assumed to conclude that this tuple <math>L</math> exists; the axiom of choice is not needed in some situations, such as when <math>I</math> is finite or when every <math>L_i \in X_i</math> is the {{em|unique}} limit of the net <math>\pi_i\left(f_\bull\right)</math> (because then there is nothing to choose between), which happens for example, when every <math>X_i</math> is a [[Hausdorff space]]. If <math>I</math> is infinite and <math>{\textstyle\prod} X_\bull = {\textstyle\prod\limits_{j \in I}} X_j</math> is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections <math>\pi_i : {\textstyle\prod} X_\bull \to X_i</math> are [[surjective map]]s.

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 (mathematics)|subnet]].

===Limit superior/inferior===

[[Limit superior]] and [[limit inferior]] of a net of real numbers can be defined in a similar manner as for sequences.<ref>Aliprantis-Border, p. 32</ref><ref>Megginson, p. 217, p. 221, Exercises 2.53–2.55</ref><ref>Beer, p. 2</ref> Some authors work even with more general structures than the real line, like complete lattices.<ref>Schechter, Sections 7.43–7.47</ref>

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 sum]]s where the net's directed set is the set of all [[partition of an interval|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 number]]s. 
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>{{sfn|Willard|2004|p=77}} 

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>).{{sfn|Willard|2004|p=77}} 

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-order|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>

The first example is a special case of this with <math>c = \omega.</math>

See also [[Order topology#Ordinal-indexed sequences|ordinal-indexed sequence]].