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diff --git a/mindmap/metric space.org b/mindmap/metric space.org index 2609691..338686c 100644 --- a/mindmap/metric space.org +++ b/mindmap/metric space.org @@ -2,14 +2,13 @@ :ID: 6f24f731-60e5-4904-88d7-c63869505981 :ROAM_ALIASES: metric :END: -#+title: metric space +#+title: Metric Space #+author: Preston Pan #+description: The basis of modern analysis. - #+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. +A metric space $(X, d)$ is a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]] with a metric $d(x,y): X \times X \rightarrow \mathbb{R}$ defined on members of the set. This metric is a generalization of distance, with the following properties: \begin{align} \label{} @@ -18,4 +17,6 @@ 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. +where property $(4)$ is the triangle inequality. Also, the metric generates the [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] on the open sets; a basis can be chosen by including every +open ball, which is defined as $B(x, r) = \lbrace y: d(x, y) < r\rbrace$. A neighbourhood basis can be chosen by including every open rational ball +that is a neighbourhood of $x$, and in fact this neighbourhood basis is countable, so metric spaces are first countable. |