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author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
commit | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (patch) | |
tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 /pages/general-topology/compactness/basics.md | |
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diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md new file mode 100644 index 0000000..a1dded7 --- /dev/null +++ b/pages/general-topology/compactness/basics.md @@ -0,0 +1,43 @@ +--- +title: Basics +parent: Compactness +grand_parent: General Topology +nav_order: 1 +published: false +# cspell:words +--- + +# {{ page.title }} of Compact Spaces + +*Compact space* is short for compact topological space. + +{: .definition } +> Suppose $X$ is a topological space. +> A *covering* of $X$ is a collection $\mathcal{A}$ +> of subsets of $X$ such that +> $\bigcup \mathcal{A} = X$. +> A covering $\mathcal{A}$ of $X$ is called *open* +> if each member of the collection $\mathcal{A}$ +> is open in $X$. +> A covering $\mathcal{A}$ is called *finite* +> the collection $\mathcal{A}$ is finite. +> A *subcovering* of a covering $\mathcal{A}$ of $X$ +> is a subcollection $\mathcal{B}$ of $\mathcal{A}$ +> such that $\mathcal{B}$ is a covering of $X$. + +{: .definition } +> A topological space $X$ is called *compact* +> if every open covering of $X$ +> has a finite subcovering. + +{: .theorem } +> Every closed subspace of a compact space is compact. + +{% proof %} +{% endproof %} + +{: .theorem } +> Every compact subspace of a Hausdorff space is closed. + +{% proof %} +{% endproof %} |