Directed Set

1. Definition
2. Nets
- 2.1. Common Definitions
  - 2.1.1. Frequently
  - 2.1.2. Eventually
- 2.2. Universal Nets
3. Pitfalls

1. Definition

A directed set $D$ is a set with some preorder defined on it:

\begin{aligned} \forall \alpha, \beta \in D, \exists \gamma, \alpha \le \gamma, \beta \le \gamma\end{aligned}

where $\ge$ obeys the usual rules for preorders (by convention, when we say $\alpha \le \gamma$ we are saying $\gamma \ge \alpha$ ). Though we will just use partial order notation because the theory is equivalent if you just factor out by some equivalence relation.

2. Nets

This notion is central to the study of compactness in the way that sequences are. A net is a function $f: D \rightarrow X$ which maps directed set elements into members of a Topological Space. There is one main theorem regarding nets that are of central importance, which is that every net has a universal subnet. This mirrors the Bolzano-Weierstrass Theorem in sequences, and has deep implications for compactness. We will give an explanation of universality as well as some definitions to aide the explanation.

2.1. Common Definitions

These are some common definitions for nets which are used in topology to define abstracted notions of convergence and compactness.

2.1.1. Frequently

A net $\lbrace x_{\alpha} \rbrace$ is frequently in some set $A$ if for all $\alpha \in D$ , there exists $\beta \in D$ such that $\beta \ge \alpha, x_{\beta} \in A$ .

2.1.2. Eventually

A net $\lbrace x_{\alpha} \rbrace$ is eventually in some set $A$ if there exists $\alpha \in D$ such that for all $\beta \ge \alpha$ , $x_{\beta}\in A$ .

Often this definition is used as a shorthand in order to

2.2. Universal Nets

Universal nets are defined as nets that are either eventually in $A$ or eventually in $A^{c}$ for all $A$ in a topological space $X$ . Clearly, they are of great importance to the study of both order theory and topology. The main theorem is this:

every net has a universal subnet.

Use Zorn's lemma or the Axiom of choice.

and can be used to prove Tychonoff's theorem, a main result in the study of compact Hausdorff Spaces.

3. Pitfalls

Note these couple facts:

subnets of sequences are not always sequences! Subnets can branch, repeat, and use entirely different directed sets. The only requirement is that subnets preserve order.
nets don't converge uniquely in general; only when the space is a Hausdorff Space do nets converge uniquely when they do converge.