From 55e29f03ac3b537843f85892a1323e1f46321675 Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Sat, 11 Apr 2026 13:14:04 -0700 Subject: new articles and snippets --- mindmap/directed set.org | 59 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 mindmap/directed set.org (limited to 'mindmap/directed set.org') diff --git a/mindmap/directed set.org b/mindmap/directed set.org new file mode 100644 index 0000000..3b3d6a1 --- /dev/null +++ b/mindmap/directed set.org @@ -0,0 +1,59 @@ +: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. -- cgit v1.3