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| author | Preston Pan <ret2pop@nullring.xyz> | 2026-04-11 14:09:26 -0700 |
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| committer | Preston Pan <ret2pop@nullring.xyz> | 2026-04-11 14:09:26 -0700 |
| commit | 56580b952901242786da04d0cb3d8e1ea1698db7 (patch) | |
| tree | dffdd0d6a5edc3c8aebca74314745a53c1176512 /mindmap/Compactness.org | |
| parent | f17203b32bd1ecb0d908bbf03b9239e2efde59d6 (diff) | |
| parent | 75cfba0e2e705ed0a87abcd5b6822beed8fad555 (diff) | |
Merge branch 'mindmap'
Diffstat (limited to 'mindmap/Compactness.org')
| -rw-r--r-- | mindmap/Compactness.org | 14 |
1 files changed, 14 insertions, 0 deletions
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. |