:PROPERTIES:
:ID:       2517cbfe-bd7b-474f-993d-d4ee3c65a069
:END:
#+title: Directed Set
#+author: Preston Pan
#+description: Central in order theory.
#+options: broken-links:t

* Definition
A directed set $D$ is a set with some preorder defined on it:
\begin{align}
  \forall \alpha, \beta \in  D, \exists \gamma,  \alpha \le \gamma, \beta \le \gamma
\end{align}
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.
* Nets
:PROPERTIES:
:ID:       d6dd23da-78be-420f-9103-4a81745aa272
:ROAM_ALIASES: net "universal net"
:END:
This notion is central to the study of compactness in the way that [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][sequences]] are. A net is a [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]]  $f: D \rightarrow X$ which maps directed set elements into
members of a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][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 [[id:1e484e9f-cfd5-48f7-a920-c242f732b452][Bolzano-Weierstrass Theorem]] in sequences, and has deep implications for [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]]. We will give an explanation of
universality as well as some definitions to aide the explanation.
** Common Definitions
These are some common definitions for nets which are used in [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] to define abstracted notions of convergence and [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]].
*** Frequently
:PROPERTIES:
:ID:       222f5770-d618-4620-8bc0-5f7c1171f417
:ROAM_ALIASES: frequently
:END:
#+begin_definition
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 $\beta \ge \alpha, x_{\beta} \in A$.
#+end_definition
*** Eventually
:PROPERTIES:
:ID:       18a8e850-963d-4cfc-810a-6568ec33b6af
:ROAM_ALIASES: eventually
:END:
#+begin_definition
A net $\lbrace x_{\alpha} \rbrace$ is /eventually/ in some set $A$ if there exists $\alpha \in D$ such that for all $\beta \ge \alpha$, $x_{\beta}\in A$.
#+end_definition
Often this definition is used as a shorthand in order to
** Universal Nets
Universal nets are defined as nets that are /either/ [[id:18a8e850-963d-4cfc-810a-6568ec33b6af][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 [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]]. The main theorem is this:
#+begin_theorem
every net has a universal subnet.
#+end_theorem

#+begin_proof
Use Zorn's lemma or the Axiom of choice.
#+end_proof
and can be used to prove Tychonoff's theorem, a main result in the study of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Spaces]].
* 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 [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Space]] do nets converge uniquely when they /do/ converge.