3 files changed, 72 insertions, 32 deletions
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 %}
diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md
index 60c29a0..37e9b4d 100644
--- a/pages/general-topology/compactness/index.md
+++ b/pages/general-topology/compactness/index.md
@@ -1,9 +1,46 @@
 ---
 title: Compactness
 parent: General Topology
-nav_order: 1
+nav_order: 5
 has_children: true
-# cspell:words
+has_toc: false
 ---
 
 # {{ page.title }}
+
+## Compactness in Terms of Closed Sets
+
+{% theorem %}
+A topological space $X$ is compact
+if and only if it has the following property:
+- Given any collection $\mathcal{C}$ of closed subsets of $X$,
+  if every finite subcollection of $\mathcal{C}$ has nonempty intersection,
+  then $\mathcal{C}$ has nonempty intersection.
+{% endtheorem %}
+
+{% proof %}
+By definition, a topological space $X$ is compact
+if and only if it has the following property:
+- Given any collection $\mathcal{O}$ of open subsets of $X$,
+  if $\mathcal{O}$ covers $X$,
+  then there exists a finite subcollection of $\mathcal{O}$ that covers $X$.
+
+If $\mathcal{A}$ is a collection of subsets of $X$,
+let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote the collection of the complements of its members.
+Clearly, $\mathcal{B}$ is a subcollection of $\mathcal{A}$
+if and only if $\mathcal{B}^c$ is a subcollection of $\mathcal{A}^c$.
+Moreover, note that $\mathcal{B}$ covers $X$ if and only if
+$\mathcal{B}^c$ has empty intersection.
+Taking the contrapositive, we reformulate above property:
+- Given any collection $\mathcal{O}$ of open subsets of $X$,
+  if every finite subcollection of $\mathcal{O}^c$ has nonempty intersection,
+  then $\mathcal{O}^c$ has nonempty intersection.
+
+To complete the proof, observe that a collection $\mathcal{A}$ consists of open subsets of $X$
+if and only if $\mathcal{A}^c$ consists of closed subsets of $X$.
+{% endproof %}
+
+{% definition Finite Intersection Property%}
+TODO
+{% enddefinition %}
+
diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md
index 2ae78e4..ddf3800 100644
--- a/pages/general-topology/compactness/tychonoff-product-theorem.md
+++ b/pages/general-topology/compactness/tychonoff-product-theorem.md
@@ -3,17 +3,17 @@ 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.
+{% theorem * Tychonoff Product Theorem %}
+The product of (an arbitrary family of) compact spaces is compact.
+{% endtheorem %}
 
 {% proof %}
 TODO
 {% endproof %}
+
+Important Application:
+[Banach–Alaoglu Theorem](/pages/functional-analysis-basics/banach-alaoglu-theorem.html).