Update

14 files changed, 547 insertions, 131 deletions

diff --git a/pages/general-topology/baire-spaces.md b/pages/general-topology/baire-spaces.md
index 6bd7d9f..1332984 100644
--- a/pages/general-topology/baire-spaces.md
+++ b/pages/general-topology/baire-spaces.md
@@ -1,25 +1,24 @@
 ---
 title: Baire Spaces
 parent: General Topology
-nav_order: 1
+nav_order: 7
 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 }
+{% definition Baire Space %}
+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.
+{% enddefinition %}
 
 Note that
 a set is dense in a topological space
@@ -33,11 +32,9 @@ 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.
+{% theorem * Baire Category Theorem #1 %}
+Every complete metric space is a Baire space.
+{% endtheorem %}
 
 {% proof %}
 Let $X$ be a metric space
@@ -76,8 +73,6 @@ 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.
+{% theorem * Baire Category Theorem #2 %}
+Every locally compact Hausdorff space is a Baire space.
+{% endtheorem %}
diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt
deleted file mode 100644
index eabe792..0000000
--- a/pages/general-topology/baire-spaces.md.txt
+++ /dev/null
@@ -1,73 +0,0 @@
----
-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
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).
diff --git a/pages/general-topology/connectedness.md b/pages/general-topology/connectedness.md
new file mode 100644
index 0000000..b422775
--- /dev/null
+++ b/pages/general-topology/connectedness.md
@@ -0,0 +1,13 @@
+---
+title: Connectedness
+parent: General Topology
+nav_order: 5
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md
new file mode 100644
index 0000000..7ae4534
--- /dev/null
+++ b/pages/general-topology/continuity-and-convergence.md
@@ -0,0 +1,24 @@
+---
+title: Continuity & Convergence
+parent: General Topology
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% definition Continuity %}
+A mapping $f: X \to Y$ between topological spaces $X$ and $Y$ is called *continuous*,
+if for each open subset $V$ of $Y$ the inverse image $f^{-1}(V)$ is an open subset of $X$.
+{% enddefinition %}
+
+Slogan: continuous $=$ The inverse image of every open subset is open.
+
+{% definition Homeomorphism %}
+Suppose $X$ and $Y$ are topological spaces.
+A mapping $f: X \to Y$ is said to be a *homeomorphism*,
+if $f$ is bijective and both $f$ and the inverse mapping $f^{-1} : Y \to X$ are continuous.
+{% enddefinition %}
+
+## Nets
+
+## Filters
diff --git a/pages/general-topology/jordan-curve-theorem.md b/pages/general-topology/jordan-curve-theorem.md
index 9da141e..e043fdc 100644
--- a/pages/general-topology/jordan-curve-theorem.md
+++ b/pages/general-topology/jordan-curve-theorem.md
@@ -3,16 +3,12 @@ title: Jordan Curve Theorem
 parent: General Topology
 nav_order: 1
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> ...
+{% theorem %}
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md
new file mode 100644
index 0000000..c0dc45a
--- /dev/null
+++ b/pages/general-topology/metric-spaces/index.md
@@ -0,0 +1,215 @@
+---
+title: Metric Spaces
+parent: General Topology
+nav_order: 8
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+{% definition Metric, Metric Space %}
+A *metric* (or *distance function*) on a set $X$ is
+a mapping $d : X \times X \to \RR$ with the properties \
+**(M1)** $\ \forall x,y \in X : d(x,y) = 0 \iff x=y \quad$ *(point separation)* \
+**(M2)** $\ \forall x,y \in X : d(x,y) = d(y,x) \quad$ *(symmetry)* \
+**(M3)** $\ \forall x,y,z \in X : d(x,z) \le d(x,y) + d(y,x) \quad$ *(triangle inequality)* \
+We say that $d(x,y)$ is the *distance* between $x$ and $y$. \
+A *metric space* is a pair $(X,d)$ consisting of a set $X$
+and a metric $d$ on $X$.
+{% enddefinition %}
+
+Setting $x=z$ in **(M3)** and applying **(M1)** & **(M2)**
+gives us $0 = d(x,x) \le 2 d(x,y)$, hence $d(x,y) \ge 0$.
+This *nonnegativity* of the metric is often part of the definition.
+
+Relaxing **(M1)** to the condition $\forall x \in X : d(x,x) = 0$
+leads to the notion of a *semi-metric*
+and that of a *semi-metric space*.
+Nonnegativity still follows as shown above.
+
+*Pseudo-metric* is usually a synonym for *semi-metric*.
+
+*Quasi-metric* refers to dropping **(M2)**.
+
+An *ultrametric* satisfies in place of **(M3)** the stronger condition
+$d(x,z) \le \max \braces{d(x,y),d(y,z)}$.
+
+{% definition Metric Subspace %}
+A *metric subspace* of a metric space $(X,d)$ is a pair $(S,d_S)$
+where $S$ is a subset of $X$ and
+$d_S$ is the restriction of $d$ to $S \times S$.
+{% enddefinition %}
+
+Clearly, a metric subspace of a metric space is itself a metric space.
+
+{% proposition %}
+Let $(X,d)$ be a (semi-)metric space.
+- For all $x,y,z \in X$ we have the *inverse triangle inequality*
+  
+  $$
+  \abs{d(x,y) - d(y,z)} \le d(x,z).
+  $$
+
+- For all $v,w,x,y \in X$ we have the *quadrilateral inequality*
+  
+  $$
+  \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y)
+  $$
+{% endproposition %}
+
+The proofs are straightforward.
+
+TODO
+- isometry
+- metric induced by a norm
+- metric product
+
+{% definition Diameter %}
+The *diameter* of a subset $S$ of a metric space $(X,d)$ is the number
+
+$$
+\diam{S} = \sup \braces{d(x,y) : x,y \in S} \in \braces{-\infty} \cup [0,\infty].
+$$
+{% enddefinition %}
+
+Note that $\diam{S} = -\infty$ iff $S = \varnothing$,
+and $\diam{S} = 0$ iff $S$ is a singleton set.
+
+{% definition Distance %}
+Suppose $(X,d)$ is a metric space.
+The *distance* from a point $x \in X$ to a subset $S \subset X$ is
+
+$$
+\dist{x,S} = \inf \braces{d(x,y) : y \in S} \in [0,\infty].
+$$
+{% enddefinition %}
+
+Note that $\dist{x,S} = \infty$ iff $S = \varnothing$.
+
+{% definition Convergence, Limit, Divergence %}
+Let $(X,d)$ be a metric space.
+A sequence $(x_n)_{n \in \NN}$ in $X$ is said to *converge to a point $x \in X$*, if
+
+$$
+\forall \epsilon > 0 \ \ \exists N \in \NN \ \ \forall n \ge N : d(x,x_n) < \epsilon.
+$$
+
+In this case, we call $x$ a *limit (point)* of the sequence.
+Symbolically this is expressed by
+
+$$
+\lim_{n \to \infty} x_n = x
+$$
+
+or by saying that $x_n \to x$ as $n \to \infty$.
+
+We call a sequence in $X$ *convergent*
+if it converges to some point of $X$
+and *divergent* otherwise.
+{% enddefinition %}
+
+For a semi-metric space the definition remains the same.
+However, the notation $\lim x_n = x$ can be misleading,
+because there might be more than one limit point.
+
+{% proposition %}
+A sequence in a metric space has at most one limit.
+{% endproposition %}
+
+In other words: The limit of a convergent sequence in a metric space is unique.
+
+{% proof %}
+Let $(x_n)$ be a convergent sequence in a metric space $(X,d)$ with limit point $x$.
+If $x'$ is another limit point of $(x_n)$,
+then $d(x,x') \le d(x,x_n) + d(x_n,x')$ for all $n \in \NN$ by **(M3)**.
+Given $\epsilon >0$, there exist natural numbers $N$ and $N'$ such that
+$d(x,x_n) < \epsilon$ for all $n \ge N$ and
+$d(x,x_n) < \epsilon$ for all $n \ge N'$.
+Both hold, if $n$ is large enough ($\ge \max \braces{N,N'}$ to be precise).
+It follows that $d(x,x') < 2 \epsilon$.
+Since $\epsilon$ was arbitrary, $d(x,x') = 0$.
+Therefore, $x=x'$ by **(M1)**.
+{% endproof %}
+
+{% corollary %}
+A semi-metric space $X$ is a metric space if and only if
+every sequence in $X$ has at most one limit.
+{% endcorollary %}
+
+{% definition %}
+Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
+A mapping $f: X \to Y$ is called
+- *continuous at a point $x \in X$* if
+  
+  $$
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *continuous* if it is continuous at every point of $X$, that is
+  
+  $$
+  \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *uniformly continuous* if
+  
+  $$
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *Lipschitz continuous* if
+  
+  $$
+  \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') 
+  $$
+{% enddefinition %}
+
+{% definition Open Ball, Closed Ball, Sphere %}
+Suppose $(X,d)$ is a metric space.
+The *open ball* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+B(c,r) = \braces{x \in X : d(x,c) < r}.
+$$
+
+The *closed ball* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+\overline{B}(c,r) = \braces{x \in X : d(x,c) \le r}.
+$$
+
+The *sphere* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+S(c,r) = \braces{x \in X : d(x,c) = r}.
+$$
+{% enddefinition %}
+
+Observe that $S(c,r) = \overline{B}(c,r) \setminus B(c,r)$.
+
+{% definition Open Subset of a Metric Space %}
+A subset $O$ of a metric space is called *open* if for every point $x \in O$
+there exists an $\epsilon > 0$ such that $B(x,\epsilon) \subset O$.
+{% enddefinition %}
+
+{% proposition Metric Topology %}
+Let $(X,d)$ be a metric space.
+The collection of open subsets of $X$ forms a topology on $X$.
+This topology is called the *metric topology* on $X$ induced by $d$.
+{% endproposition %}
+
+{% proposition %}
+- Open balls in a metric space are open with respect to the metric topology.
+- Closed balls in a metric space are closed with respect to the metric topology.
+- The boundary (with respect to the metric topology) of an open or closed ball
+  is the sphere with the same center and radius. Not true!!!!
+- The collection of open balls in a metric space forms a basis of the metric topology.
+{% endproposition %}
+
+## Complete Metric Spaces
+
+- Definition
+- Banach Fixed-Point Theorem
+- Baire
+- Metric Completion
diff --git a/pages/general-topology/separation/index.md b/pages/general-topology/separation/index.md
new file mode 100644
index 0000000..b4916f5
--- /dev/null
+++ b/pages/general-topology/separation/index.md
@@ -0,0 +1,9 @@
+---
+title: Separation
+parent: General Topology
+nav_order: 4
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/general-topology/separation/tietze-extension-theorem.md b/pages/general-topology/separation/tietze-extension-theorem.md
new file mode 100644
index 0000000..4b2eee3
--- /dev/null
+++ b/pages/general-topology/separation/tietze-extension-theorem.md
@@ -0,0 +1,14 @@
+---
+title: Tietze Extension Theorem
+parent: Separation
+grand_parent: General Topology
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/separation/urysohn-lemma.md b/pages/general-topology/separation/urysohn-lemma.md
new file mode 100644
index 0000000..b3f8208
--- /dev/null
+++ b/pages/general-topology/separation/urysohn-lemma.md
@@ -0,0 +1,14 @@
+---
+title: Urysohn Lemma
+parent: Separation
+grand_parent: General Topology
+nav_order: 1
+---
+
+# The {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md
new file mode 100644
index 0000000..b0b1834
--- /dev/null
+++ b/pages/general-topology/topological-spaces.md
@@ -0,0 +1,146 @@
+---
+title: Topological Spaces
+parent: General Topology
+nav_order: 1
+---
+
+# {{ page.title }}
+
+## Elementary Concepts
+
+{% definition Topology, Topological Space %}
+A *topology* on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that \
+**(T1)** $\varnothing$ and $X$ belong to $\mathcal{T}$, \
+**(T2)** the union of any subcollection of $\mathcal{T}$ belongs to $\mathcal{T}$, \
+**(T3)** the intersection of any finite subcollection $\mathcal{T}$ belongs to $\mathcal{T}$. \
+A *topological space* is a pair $(X,\mathcal{T})$ consisting of
+a set $X$ and a topology $\mathcal{T}$ on $X$.
+{% enddefinition %}
+
+If one follows the convention that
+the union of the empty collection of subsets of $X$ is the empty subset of $X$,
+and its intersection is all of $X$,
+then **(T1)** is a consequence of **(T2)**, **(T3)**
+and can be omitted.
+
+If $(X,\mathcal{T})$ is a topological space,
+the elements of $X$ are called *points*
+and the elements of $\mathcal{T}$ are called the *open sets*.
+
+{% example %}
+On every set $X$ we have
+the *trivial* (or *indiscrete*) *topology* $\braces{\varnothing,X}$ and
+the *discrete topology* $\mathcal{P}(X)$.
+These collections are in fact topologies on $X$.
+{% endexample %}
+
+{% example %}
+If $X$ is any set,
+then the collection of all subsets of $X$
+whose complement is either finite or all of $X$
+is a topology on $X$;
+it is called the *finite complement topology*.
+The *countable complement topology* is defined analogously.
+{% endexample %}
+
+{% definition Comparison of Topologies %}
+Suppose $\mathcal{T}$ and $\mathcal{T}'$ are topologies on a set $X$.
+When $\mathcal{T} \subset \mathcal{T}'$,
+we say that $\mathcal{T}$ is *coarser* or *smaller* or *weaker* than $\mathcal{T}'$,
+and that $\mathcal{T}'$ is *finer* or *larger* or *stronger* than $\mathcal{T}$.
+If the inclusion is proper, then we say *strictly coarser* and so on.
+If either $\mathcal{T} \subset \mathcal{T}'$ or $\mathcal{T} \supset \mathcal{T}'$ holds,
+then the topologies are said to be *comparable*.
+{% enddefinition %}
+
+{% proposition Intersection of Topologies %}
+If $\braces{\mathcal{T}_{\alpha}}$ is a family of topologies on a set $X$,
+then $\bigcap_{\alpha} \mathcal{T}_{\alpha}$ is a topology on $X$.
+{% endproposition %}
+
+{% definition Generated Topology %}
+Suppose $\mathcal{A}$ is a collection of subsets of a set $X$.
+The *topology generated by $\mathcal{A}$* is
+the intersection of all topologies on $X$ containing $\mathcal{A}$.
+{% enddefinition %}
+
+By the previous proposition, the generated topology is indeed a topology.
+
+{% proposition %}
+The topology generated by a collection $\mathcal{A}$ of subsets of a set $X$
+is the smallest topology on $X$ containing $\mathcal{A}$.
+{% endproposition %}
+
+## Bases and Subbases
+
+{% definition Basis for a Topology %}
+A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$
+such that for every point $x \in X$ 
+- there exists $B \in \mathcal{B}$ such that $x \in B$,
+- if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$,
+  then there exists a $B_3 \in \mathcal{B}$
+  such that $x \in B_3 \subset B_1 \cap B_2$.
+{% enddefinition %}
+
+{% theorem Topology Generated by a Basis %}
+If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$,
+then the topology generated by $\mathcal{B}$ equals
+- the collection of all subsets $S \subset X$ with the property
+  that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$
+  such that $x \in B$ and $B \subset S$;
+- the collection of all arbitrary unions of elements of $\mathcal{B}$.
+{% endtheorem %}
+
+Let $\mathcal{T}$ be a topology on a set $X$.
+As one might expect,
+a collection $\mathcal{B}$ of subsets of $X$
+is said to be a *basis for the topology $\mathcal{T}$*,
+if $\mathcal{B}$ is basis for a topology on $X$ and
+the topology generated by $\mathcal{B}$ equals $\mathcal{T}$.
+
+{% example %}
+If $X$ is a set, then the collection of singletons $\braces{x}$, $x \in X$,
+is a basis for the discrete topology on $X$.
+{% endexample %}
+
+{% example %}
+If $(X,d)$ is a metric space,
+then the collection of open balls is a basis for the metric topology on $X$.
+{% endexample %}
+
+{% definition Subbasis for a Topology %}
+A *subbasis for a topology* on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$
+such that for every point $x \in X$ there exists a $S \in \mathcal{S}$ such that $x \in S$.
+{% enddefinition %}
+
+{% theorem Topology Generated by a Subbasis %}
+If $X$ is set and $\mathcal{S}$ is a subbasis for a topology on $X$,
+then the topology generated by $\mathcal{S}$ equals
+- the collection of all arbitrary unions of finite intersections of elements of $\mathcal{S}$.
+{% endtheorem %}
+
+## Open and Closed Sets
+
+{% definition Open Set, Closed Set %}
+Suppose $(X,\mathcal{T})$ is a topological space.
+A subset $S$ of $X$
+is called *open* with respect to $\mathcal{T}$
+when it belongs to $\mathcal{T}$
+and it is called *closed* with respect to $\mathcal{T}$
+when its complement $X \setminus S$ belongs to $\mathcal{T}$.
+{% enddefinition %}
+
+A subset of a topological space is open
+if and only if its complement is closed.
+
+{% proposition %}
+Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then
+{: .mb-0 }
+- $X$ and $\varnothing$ belong to $\mathcal{C}$,
+-  the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$,
+-  the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$.
+{% endproposition %}
+
+## The Subspace Topology
+
+
diff --git a/pages/general-topology/universal-constructions.md b/pages/general-topology/universal-constructions.md
new file mode 100644
index 0000000..827f730
--- /dev/null
+++ b/pages/general-topology/universal-constructions.md
@@ -0,0 +1,23 @@
+---
+title: Universal Constructions
+parent: General Topology
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Initial Topology %}
+Suppose that $f_i : S \to X_i$, $i \in I$, is a family of maps,
+from a set $S$ into topological spaces $X_i$.
+The *initial topology* on $S$ induced by the family $(f_i)$
+is defined to be the weakest topology on $S$
+making all maps $f_i$ continuous.
+{% enddefinition %}
+
+{% theorem * Universal Property of the Initial Topology %}
+The initial topology on $S$ induced by the family $(f_i)$
+is the unique topology on $S$ with the property that
+for any topological space $T$,
+a mapping $g : T \to S$ is continuous if and only if
+all compositions $f_i \circ g : T \to X_i$ are continuous.
+{% endtheorem %}