From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

---
 pages/general-topology/compactness/basics.md | 51 +++++++++++++++-------------
 1 file changed, 27 insertions(+), 24 deletions(-)

(limited to 'pages/general-topology/compactness/basics.md')

diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md
index a1dded7..c7249ff 100644
--- a/pages/general-topology/compactness/basics.md
+++ b/pages/general-topology/compactness/basics.md
@@ -4,40 +4,43 @@ 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.
+{% 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 $X$.
+{% enddefinition %}
+
+{% definition %}
+A topological space $X$ is called *compact*
+if every open covering of $X$
+has a finite subcovering.
+{% enddefinition %}
+
+{% theorem %}
+Every closed subspace of a compact space is compact.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .theorem }
-> Every compact subspace of a Hausdorff space is closed.
+{% theorem %}
+Every compact subspace of a Hausdorff space is closed.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
-- 
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