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 $x\in \overline{A}$ (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 $a\in \overline{A}$ if and only if there is a net in $A$ that converges to $a$.