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diff --git a/mindmap/Compactness.org b/mindmap/Compactness.org new file mode 100644 index 0000000..e175c43 --- /dev/null +++ b/mindmap/Compactness.org @@ -0,0 +1,14 @@ +:PROPERTIES: +:ID: 72deb4cd-46f7-4ef2-9c66-6943e47a9e83 +:ROAM_ALIASES: "open cover" compact compactness +:END: +#+title: Compactness +#+author: Preston Pan +#+description: Basic analysis and topology. +#+options: broken-links:t +* Introduction +A compact [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]] is a topological space such that every open cover has a finite subcover. That is, if $\mathbb{U}$ is a collection of open +sets $U$ that cover $X$, then there exists a subset $V$ of $\mathbb{U}$ such that $V$ is finite and covers $X$. + +An equivalent definition is that of in terms of [[id:d6dd23da-78be-420f-9103-4a81745aa272][nets]]; a set is compact if and only if all [[id:d6dd23da-78be-420f-9103-4a81745aa272][universal nets]] converge. We will prove this in this article, +as well as several basic properties and definitions related to compactness. |