Directed Sets[edit | edit source]
Directed sets are very useful in topology. We will explore a couple of their uses in this lesson.
Definition[edit | edit source]
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)[edit | edit source]
- Let be a set. Then its power set is a directed set, ordered by set inclusion. Indeed, if then and .
- 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[edit | edit source]
Let be a directed set. A subset is cofinal if for every there is such that .
Examples (cofinal set)[edit | edit source]
- 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.
- 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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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.
- If a set is compact, then every sequence in it has a convergent subsequence.
- If a function is continuous and then .
- 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.
- 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).
- 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 ).
- 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[edit | edit source]
Prove each of the following.
- A set is compact if and only if every net in has a convergent subnet.
- A function is continuous if and only if whenever is a net converging to .
- Let . Then if and only if there is a net in that converges to .