Editing Net (mathematics) (section)

==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 (topology)|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>.{{sfn|Willard|2004|p=75}} 

=== Open sets and characterizations of topologies ===

{{See also|Axiomatic foundations of topological spaces#Definition via convergence of nets}}

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>{{sfn|Howes|1995|pp=83–92}} 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 [[Topology (structure)|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 function (topology)|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>{{sfn|Willard|2004|p=75}} 
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]]).

{{collapse top|title=Proof|left=true}}
(<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 (topology)|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 [[Neighbourhood (mathematics)|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 [[Subset|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. 
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=== 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]].

{{collapse top|title=Proof|left=true}}
(<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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===Cluster and limit points===

The set of cluster points of a net is equal to the set of limits of its convergent [[Subnet (mathematics)|subnet]]s.

{{collapse top|title=Proof|left=true}}
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>
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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 {{em|equivalent}} 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.