7 files changed, 256 insertions, 0 deletions
diff --git a/pages/general-topology/baire-spaces.md b/pages/general-topology/baire-spaces.md
new file mode 100644
index 0000000..6bd7d9f
--- /dev/null
+++ b/pages/general-topology/baire-spaces.md
@@ -0,0 +1,83 @@
+---
+title: Baire Spaces
+parent: General Topology
+nav_order: 1
+description: >
+  A Baire space is a topological space with the property that the intersection
+  of countably many dense open subsets is still dense. One version of the Baire
+  Category Theorem states that complete metric spaces are Baire spaces. We give
+  a self-contained proof of Baire's Category Theorem by contradiction.
+# spellchecker:words
+---
+
+# {{ page.title }}
+
+{: .definition }
+> A topological space is said to be a *Baire space*,
+> if any of the following equivalent conditions holds:
+> {: .mb-0 }
+>
+> - The intersection of countably many dense open subsets is still dense.
+> - The union of countably many closed subsets with empty interior has empty interior.
+> {: .mt-0 .mb-0 }
+
+Note that
+a set is dense in a topological space
+if and only if
+its complement has empty interior.
+
+Any sufficient condition
+for a topological space to be a Baire space
+constitutes a *Baire Category Theorem*,
+of which there are several.
+Here we give one
+that is commonly used in functional analysis.
+
+{: .theorem-title }
+> Baire Category Theorem #1
+> {: #baire-category-theorem }
+>
+> Every complete metric space is a Baire space.
+
+{% proof %}
+Let $X$ be a metric space
+with complete metric $d$.
+Suppose that $X$ is not a Baire space.
+Then there is a countable collection $\braces{U_n}$ of dense open subsets of $X$
+such that the intersection $U := \bigcap U_n$ is not dense in $X$.
+
+In a metric space, any nonempty open set contains an open ball.
+It is also true, that any nonempty open set contains a closed ball,
+since $\overline{B(y,\delta_1)} \subset B(y,\delta_2)$ if $\delta_1 < \delta_2$.
+
+We construct a sequence $(B_n)$ of open balls $B_n := B(x_n,\epsilon_n)$ satisfying
+
+$$
+\overline{B_{n+1}} \subset B_n \cap U_n \quad \epsilon_n < \tfrac{1}{n} \quad \forall n \in \NN,
+$$
+
+as follows: By hypothesis,
+the interior of $X \setminus U$ is not empty (otherwise $U$ would be dense in $X$),
+so we may choose an open ball $B_1$ with $\epsilon_1 < 1$
+such that $\overline{B_1} \subset X \setminus U$.
+Given $B_n$,
+the set $B_n \cap U_n$ is nonempty, because $U_n$ is dense in $X$,
+and it is open, because $B_n$ and $U_n$ are open.
+This allows us to choose an open ball $B_{n+1}$ as desired.
+
+Note that by construction $B_m \subset B_n$ for $m \ge n$,
+thus $d(x_m,x_n) < \epsilon_n < \tfrac{1}{n}$.
+Therefore, the sequence $(x_n)$ is Cauchy
+and has a limit point $x$ by completeness.
+In the limit $m \to \infty$, we obtain $d(x,x_n) \le \epsilon_n$ (strictness is lost),
+hence $x \in \overline{B_n}$ for all $n$.
+This shows that $x \in U_n$ for all $n$, that is $x \in U$.
+On the other hand, $x \in \overline{B_1} \subset X \setminus U$,
+in contradiction to the preceding statement.
+{% endproof %}
+
+{: .theorem-title }
+> Baire Category Theorem #2
+> {: #baire-category-theorem }
+>
+> Every compact Hausdorff space is a Baire space.
diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt
new file mode 100644
index 0000000..eabe792
--- /dev/null
+++ b/pages/general-topology/baire-spaces.md.txt
@@ -0,0 +1,73 @@
+---
+title: Baire Spaces
+parent: General Topology
+nav_order: 1
+description: >
+  A Baire space is a topological space with the property that the intersection
+  of countably many dense open subsets is still dense. One version of the Baire
+  Category Theorem states that complete metric spaces are Baire spaces. We give
+  a self-contained proof of Baire's Category Theorem by contradiction.
+# spellchecker:words
+---
+
+# 
+
+
+A topological space is said to be a *Baire space*,
+if any of the following equivalent conditions holds:
+>
+- The intersection of countably many dense open subsets is still dense.
+- The union of countably many closed subsets with empty interior has empty interior.
+
+
+Note that
+a set is dense in a topological space
+if and only if
+its complement has empty interior.
+
+Any sufficient condition
+for a topological space to be a Baire space
+constitutes a *Baire Category Theorem*,
+of which there are several.
+Here we give one
+that is commonly used in functional analysis.
+
+
+Baire Category Theorem
+
+>
+Complete metric spaces are Baire spaces.
+
+**Proof:**
+Let C-C-C be a metric space
+with complete metric D-D-D.
+Suppose that F-F-F is not a Baire space.
+Then there is a countable collection G-G-G of dense open subsets of B-B-B
+such that the intersection C-C-C is not dense in D-D-D.
+
+In a metric space, any nonempty open set contains an open ball.
+It is also true, that any nonempty open set contains a closed ball,
+since F-F-F if G-G-G.
+
+We construct a sequence B-B-B of open balls C-C-C satisfying
+
+V-V-V
+as follows: By hypothesis,
+the interior of D-D-D is not empty (otherwise F-F-F would be dense in G-G-G),
+so we may choose an open ball B-B-B with C-C-C
+such that D-D-D.
+Given F-F-F,
+the set G-G-G is nonempty, because B-B-B is dense in C-C-C,
+and it is open, because D-D-D and F-F-F are open.
+This allows us to choose an open ball G-G-G as desired.
+
+Note that by construction B-B-B for C-C-C,
+thus D-D-D.
+Therefore, the sequence F-F-F is Cauchy
+and has a limit point G-G-G by completeness.
+In the limit B-B-B, we obtain C-C-C (strictness is lost),
+hence D-D-D for all F-F-F.
+This shows that G-G-G for all B-B-B, that is C-C-C.
+On the other hand, D-D-D,
+in contradiction to the preceding statement.
+
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 %}
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
@@ -0,0 +1,9 @@
+---
+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
--- /dev/null
+++ 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 %}
diff --git a/pages/general-topology/index.md b/pages/general-topology/index.md
new file mode 100644
index 0000000..507c29a
--- /dev/null
+++ b/pages/general-topology/index.md
@@ -0,0 +1,11 @@
+---
+title: General Topology
+nav_order: 1
+has_children: true
+---
+
+# {{ page.title }}
+
+## Recommended Textbooks
+
+{% bibliography --file general-topology %}
diff --git a/pages/general-topology/jordan-curve-theorem.md b/pages/general-topology/jordan-curve-theorem.md
new file mode 100644
index 0000000..9da141e
--- /dev/null
+++ b/pages/general-topology/jordan-curve-theorem.md
@@ -0,0 +1,18 @@
+---
+title: Jordan Curve Theorem
+parent: General Topology
+nav_order: 1
+published: false
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem-title }
+> {{ page.title }}
+> {: #{{ page.title | slugify }} }
+>
+> ...
+
+{% proof %}
+{% endproof %}