Topology/Lesson 5

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

Definition
A directed set is a set $$D$$ with a partial order denoted by $$\le$$ which satisfies the additional requirement that given $$a,b\in D$$ there is $$c\in D$$ such that $$a\le c$$ and $$b\le c$$.

Examples (directed set)

 * 1) Let $$X$$ be a set.  Then its power set $$2^X$$ is a directed set, ordered by set inclusion.  Indeed, if $$A,B\subset X$$ then $$A\subset A\cup B$$ and $$B\subset A\cup B$$.
 * 2) Suppose that $$X$$ is a topological space and $$x\in X$$.  Then the set $$D$$ of all neighborhoods of $$x$$ is a directed set, ordered by reverse set inclusion (that is, $$A\le B$$ if $$A\supset B$$).  The proof is left as an exercise.

Cofinal set
Let $$D$$ be a directed set. A subset $$D^\prime\subset D$$ is cofinal if for every $$d\in D$$ there is $$d^\prime\in D^\prime$$ such that $$d\le d^\prime$$.

Examples (cofinal set)

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

Nets
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 $$\mathbb{N}$$). That is, a net in a space $$X$$ is a function $$f:D\to X$$, where $$D$$ is a directed set.

Subnet
Let $$f:D\to X$$ be a net. A subnet of $$f$$ is the restriction of $$f$$ to a subset $$D^\prime\subset D$$ that is also directed and is cofinal in $$D$$.

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
Let $$f:D\to X$$ be a net. The net converges to a point $$x\in X$$ if for every neighborhood $$N\ni x$$, there is $$a\in D$$ such that $$f(\alpha)\in N$$ for all $$\alpha\ge a$$.

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 $$D$$, not all points are assumed to be comparable (that is, there might be $$a,b\in D$$ for which neither $$a\le b$$ nor $$b\le a$$ is true). Therefore, the quantifier "for all $$\alpha\ge a$$" excludes any point in $$D$$ that is not comparable to $$a$$.

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 $$C$$ is compact, then every sequence in it has a convergent subsequence.
 * 2) If a function $$f:X\to Y$$ is continuous and $$x_n\to x$$ then $$f(x_n)\to f(x)$$.
 * 3) Let $$\{x_n\}$$ be a sequence in a set $$A\subset X$$.  If $$x_n\to x$$ in $$X$$ then $$x\in \bar A$$ (the closure of $$A$$).

Each of these is a very good result. However, for each one the converse is false. Consider the following examples.
 * 1) The space $$\omega_1$$ (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 $$\omega_1$$ is convergent).
 * 2) Let $$f:[0,\omega_1]\to \{0,1\}$$ be defined by $$f(\alpha)=0$$ for $$\alpha<\omega_1>$$ and $$f(\omega_1)=1$$.  $$[0,\omega_1]$$ has the order topology, since it is an ordinal, and $$\{0,1\}$$ has the discrete topology.  Then $$f$$ is not continuous but every sequence in $$[0,\omega_1]$$ is preserved (that is, if $$x_n\to x$$ in $$[0,\omega_1]$$ then $$f(x_n)\to f(x)$$).
 * 3) The point $$\omega_1$$ in the space $$[0,\omega_1]$$ (as in the previous example) is in the closure of $$[0,\omega_1)$$ 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
Prove each of the following.
 * 1) A set $$C$$ is compact if and only if every net in $$C$$ has a convergent subnet.
 * 2) A function $$f:X\to Y$$ is continuous if and only if $$f(\nu(\alpha))\to f(x_0)$$ whenever $$\nu:D\to X$$ is a net converging to $$x_0\in X$$.
 * 3) Let $$A\subset X$$.  Then $$a\in \bar A$$ if and only if there is a net in $$A$$ that converges to $$a$$.