Directed Sets

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Directed sets are very useful in topology. We will explore a couple of their uses in this lesson.

Definition

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A directed set is a set   with a partial order denoted by   which satisfies the additional requirement that given   there is   such that   and  .

Examples (directed set)

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  1. Let   be a set. Then its power set   is a directed set, ordered by set inclusion. Indeed, if   then   and  .
  2. Suppose that   is a topological space and  . Then the set   of all neighborhoods of   is a directed set, ordered by reverse set inclusion (that is,   if  ). The proof is left as an exercise.

Cofinal set

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Let   be a directed set. A subset   is cofinal if for every   there is   such that  .

Examples (cofinal set)

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  1. Let   be an infinite set. Then, as above, its power set   is a directed set. The subset consisting of only infinite subsets of   is a cofinal set.
  2. As in Example 2 above, let   be the set of neighborhoods of the point  . Then the set of open neighborhoods of   is a cofinal set. If   is Hausdorff and locally compact, then the set of compact neighborhoods of   is also cofinal.

Nets

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One of the main applications of directed sets is that of nets. A net is kind of like a sequence, but the indexing set is a directed set rather than an ordinal set (or, specifically the set  ). That is, a net in a space   is a function  , where   is a directed set.

Subnet

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Let   be a net. A subnet of   is the restriction of   to a subset   that is also directed and is cofinal in  .

Nets are like sequences. Just as you can picture a sequence being a bunch of points in a space, and you usually think of that sequences limiting on some particular point, you can think of nets as a bunch of points in a space. And, just like sequences, nets are useful when they accumulate at a specific point (or multiple points).


Limits

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Let   be a net. The net converges to a point   if for every neighborhood  , there is   such that   for all  .

This definition looks surprisingly similar to the definition of the limit of a sequence, and it is very similar. However, note one significant difference. In  , not all points are assumed to be comparable (that is, there might be   for which neither   nor   is true). Therefore, the quantifier "for all  " excludes any point in   that is not comparable to  .

What's all the hype about? Why did topologists even invent the concept of a net? Consider the following results, prior to nets.

  1. If a set   is compact, then every sequence in it has a convergent subsequence.
  2. If a function   is continuous and   then  .
  3. Let   be a sequence in a set  . If   in   then   (the closure of  ).

Each of these is a very good result. However, for each one the converse is false. Consider the following examples.

  1. The space   (the ordinal space, which consists of all finite/countable ordinals) is not compact but every sequence in it has a convergent subsequence (in particular, every monotone sequence in   is convergent).
  2. Let   be defined by   for   and  .   has the order topology, since it is an ordinal, and   has the discrete topology. Then   is not continuous but every sequence in   is preserved (that is, if   in   then  ).
  3. The point   in the space   (as in the previous example) is in the closure of   but is not the limit of any sequence in that set.

However, if we use nets instead of sequences, each of these results becomes a biconditional. The proof of each will be left as an exercise to the student. A suggestion for each would be to follow a proof of the case where only sequences are considered.

Exercises

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Prove each of the following.

  1. A set   is compact if and only if every net in   has a convergent subnet.
  2. A function   is continuous if and only if   whenever   is a net converging to  .
  3. Let  . Then   if and only if there is a net in   that converges to  .