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/Tychonoff's Theorem.org | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) create mode 100644 mindmap/Tychonoff's Theorem.org (limited to 'mindmap/Tychonoff's Theorem.org') diff --git a/mindmap/Tychonoff's Theorem.org b/mindmap/Tychonoff's Theorem.org new file mode 100644 index 0000000..af12d8b --- /dev/null +++ b/mindmap/Tychonoff's Theorem.org @@ -0,0 +1,24 @@ +:PROPERTIES: +:ID: 80901a90-7ffd-4b86-9619-c8a71f4a2a72 +:END: +#+title: Tychonoff's Theorem +#+author: Preston Pan +#+description: +#+options: broken-links:t +* Introduction +Tychonoff's theorem is of great importance when dealing with the study of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]], and has far reaching results in the construction of the +[[id:14bebb09-2e38-4b55-adc0-97ba571331af][Stone-Cech Compactification]], notable for its universal property. +#+begin_theorem +The product of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topological spaces]] is compact. +#+end_theorem + +#+begin_proof +Let $X = \prod_{\alpha \in A} X_{\alpha}$ be a Tychonoff space. We will use the [[id:d6dd23da-78be-420f-9103-4a81745aa272][universal nets]] definition of compactness to prove $X$ is compact. +Also, we use the fact that the [[id:d6dd23da-78be-420f-9103-4a81745aa272][net]] $\lbrace x_{\beta} \rbrace$ converges in $X$ iff each of its projections $\pi_{\alpha} (x_{\beta})$ converges. + +If $f$ is a continuous mapping and $\lbrace x_{\beta} \rbrace$ is a universal net, then $f(x_{\beta})$ is universal. Therefore, because $\pi_{\alpha}$ is continuous for all +$\alpha$, and beacuse $X$ is compact for all $\alpha$, we conclude that for all universal nets $\lbrace x_{\beta} \rbrace$, the projections $\pi_{\alpha}(x_{\beta})$ converge for +all $\alpha$, and thus $\lbrace x_{\beta} \rbrace$ converges. +#+end_proof +Note that we are proving that the product of an /arbitrary family/ of compact spaces is compact, which makes the task seem a lot less difficult than it +is. Still, universal nets make the proof nice and easy. A special case is where all $X_{\alpha} = [0, 1]$. -- cgit v1.3