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diff --git a/mindmap/metric space.org b/mindmap/metric space.org new file mode 100644 index 0000000..df750db --- /dev/null +++ b/mindmap/metric space.org @@ -0,0 +1,22 @@ +:PROPERTIES: +:ID: 6f24f731-60e5-4904-88d7-c63869505981 +:ROAM_ALIASES: metric +:END: +#+title: metric space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +A metric space $(G, d)$ is a set with a metric $d(x,y): G \times G \rightarrow \mathbb{R}$ defined on members of the set. +This metric is a generalization of distance, with the following properties: +\begin{align} +\label{} +d(x, x) = 0 \\ +x \ne y \implies d(x, y) > 0 \\ +d(x, y) = d(y, x) \\ +d(x, z) \le d(x, y) + d(x, z) +\end{align} +where property $(4)$ is the triangle inequality. |