# 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 ${\displaystyle D}$ with a partial order denoted by ${\displaystyle \leq }$ which satisfies the additional requirement that given ${\displaystyle a,b\in D}$ there is ${\displaystyle c\in D}$ such that ${\displaystyle a\leq c}$ and ${\displaystyle b\leq c}$.

### Examples (directed set)

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

### Cofinal set

Let ${\displaystyle D}$ be a directed set. A subset ${\displaystyle D^{\prime }\subset D}$ is cofinal if for every ${\displaystyle d\in D}$ there is ${\displaystyle d^{\prime }\in D^{\prime }}$ such that ${\displaystyle d\leq d^{\prime }}$.

### Examples (cofinal set)

1. Let ${\displaystyle X}$ be an infinite set. Then, as above, its power set ${\displaystyle 2^{X}}$ is a directed set. The subset consisting of only infinite subsets of ${\displaystyle X}$ is a cofinal set.
2. As in Example 2 above, let ${\displaystyle D}$ be the set of neighborhoods of the point ${\displaystyle x\in X}$. Then the set of open neighborhoods of ${\displaystyle x}$ is a cofinal set. If ${\displaystyle X}$ is Hausdorff and locally compact, then the set of compact neighborhoods of ${\displaystyle x}$ 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 ${\displaystyle \mathbb {N} }$). That is, a net in a space ${\displaystyle X}$ is a function ${\displaystyle f:D\to X}$, where ${\displaystyle D}$ is a directed set.

### Subnet

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

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

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

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

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

1. The space ${\displaystyle \omega _{1}}$ (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 ${\displaystyle \omega _{1}}$ is convergent).
2. Let ${\displaystyle f:[0,\omega _{1}]\to \{0,1\}}$ be defined by ${\displaystyle f(\alpha )=0}$ for ${\displaystyle \alpha <\omega _{1}>}$ and ${\displaystyle f(\omega _{1})=1}$. ${\displaystyle [0,\omega _{1}]}$ has the order topology, since it is an ordinal, and ${\displaystyle \{0,1\}}$ has the discrete topology. Then ${\displaystyle f}$ is not continuous but every sequence in ${\displaystyle [0,\omega _{1}]}$ is preserved (that is, if ${\displaystyle x_{n}\to x}$ in ${\displaystyle [0,\omega _{1}]}$ then ${\displaystyle f(x_{n})\to f(x)}$).
3. The point ${\displaystyle \omega _{1}}$ in the space ${\displaystyle [0,\omega _{1}]}$ (as in the previous example) is in the closure of ${\displaystyle [0,\omega _{1})}$ 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 ${\displaystyle C}$ is compact if and only if every net in ${\displaystyle C}$ has a convergent subnet.
2. A function ${\displaystyle f:X\to Y}$ is continuous if and only if ${\displaystyle f(\nu (\alpha ))\to f(x_{0})}$ whenever ${\displaystyle \nu :D\to X}$ is a net converging to ${\displaystyle x_{0}\in X}$.
3. Let ${\displaystyle A\subset X}$. Then ${\displaystyle a\in {\bar {A}}}$ if and only if there is a net in ${\displaystyle A}$ that converges to ${\displaystyle a}$.