From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

---
 pages/general-topology/compactness/basics.md       | 43 ++++++++++++++++++++++
 pages/general-topology/compactness/index.md        |  9 +++++
 .../compactness/tychonoff-product-theorem.md       | 19 ++++++++++
 3 files changed, 71 insertions(+)
 create mode 100644 pages/general-topology/compactness/basics.md
 create mode 100644 pages/general-topology/compactness/index.md
 create mode 100644 pages/general-topology/compactness/tychonoff-product-theorem.md

(limited to 'pages/general-topology/compactness')

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
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+---
+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 %}
diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md
new file mode 100644
index 0000000..60c29a0
--- /dev/null
+++ b/pages/general-topology/compactness/index.md
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+---
+title: Compactness
+parent: General Topology
+nav_order: 1
+has_children: true
+# cspell:words
+---
+
+# {{ page.title }}
diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md
new file mode 100644
index 0000000..2ae78e4
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+++ b/pages/general-topology/compactness/tychonoff-product-theorem.md
@@ -0,0 +1,19 @@
+---
+title: Tychonoff Product Theorem
+parent: Compactness
+grand_parent: General Topology
+nav_order: 2
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem-title }
+> {{ page.title }}
+> {: #{{ page.title | slugify }} }
+>
+> The product of (an arbitrary family of) compact spaces is compact.
+
+{% proof %}
+TODO
+{% endproof %}
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