path: root/pages/general-topology/metric-spaces/index.md

---
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