Editing Net (mathematics) (section)

=== Compactness ===

A space <math>X</math> is [[Compact space|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]].

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(<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.
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