Editing Net (mathematics) (section)

===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>