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+---
+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