Metric Spaces

Definition (Metric, Metric Space)

A metric (or distance function) on a set $X$ is a mapping $d:X\times X\to \mathbb{R}$ with the properties
(M1) $\forall x,y\in X:d(x,y)=0⟺x=y$ (point separation)
(M2) $\forall x,y\in X:d(x,y)=d(y,x)$ (symmetry)
(M3) $\forall x,y,z\in X:d(x,z)\le d(x,y)+d(y,x)$ (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 $X$.

Setting $x=z$ in (M3) and applying (M1) & (M2) gives us $0=d(x,x)\le 2d(x,y)$, 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 $d(x,z)\le \max \{d(x,y),d(y,z)\}$.

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 $S\times S$.

Clearly, a metric subspace of a metric space is itself a metric space.

Let $(X,d)$ be a (semi-)metric space.

• For all $x,y,z\in X$ we have the inverse triangle inequality

  $$|d(x,y)-d(y,z)|\le d(x,z).$$

• For all $v,w,x,y\in X$ we have the quadrilateral inequality

  $$|d(v,w)-d(x,y)|\le d(v,x)+d(w,y)$$

The proofs are straightforward.

Definition (Isometry)

Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. We say that a mapping $f:X\to Y$ is isometric or an isometry if it obeys $d_Y(f(x),f(x^{\prime }))=d_X(x,x^{\prime })$ for all $x,x^{\prime }\in X$.

As a consequence of (M1), every isometry is injective.

TODO

• metric induced by a norm
• metric product

Definition (Diameter)

The diameter of a subset $S$ of a metric space $(X,d)$ is the number

$$\mathrm{diam}(S)=\sup \{d(x,y):x,y\in S\}\in \{-\infty \}\cup [0,\infty ].$$

Note that $\mathrm{diam}(S)=-\infty$ iff $S=\varnothing$, and $\mathrm{diam}(S)=0$ 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

$$\mathrm{dist}(x,S)=\inf \{d(x,y):y\in S\}\in [0,\infty ].$$

Note that $\mathrm{dist}(x,S)=\infty$ iff $S=\varnothing$.

Definition (Convergence, Limit, Divergence)

Let $(X,d)$ be a metric space. A sequence $(x_n{)}_{n\in \mathbb{N}}$ in $X$ is said to converge to a point $x\in X$, if

$$\forall ϵ>0\exists N\in \mathbb{N}\forall n\ge N:d(x,x_n)<ϵ.$$

In this case, we call $x$ a limit (point) of the sequence. Symbolically this is expressed by

$$\underset{n\to \infty }{\lim }{x}_n=x$$

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.

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.

A sequence in a metric space has at most one limit.

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 $x^{\prime }$ is another limit point of $(x_n)$, then $d(x,x^{\prime })\le d(x,x_n)+d(x_n,x^{\prime })$ for all $n\in \mathbb{N}$ by (M3). Given $ϵ>0$, there exist natural numbers $N$ and $N^{\prime }$ such that $d(x,x_n)<ϵ$ for all $n\ge N$ and $d(x,x_n)<ϵ$ for all $n\ge N^{\prime }$. Both hold, if $n$ is large enough ($\ge \max \{N,N^{\prime }\}$ to be precise). It follows that $d(x,x^{\prime })<2ϵ$. Since $ϵ$ was arbitrary, $d(x,x^{\prime })=0$. Therefore, $x=x^{\prime }$ by (M1). $\square$

A semi-metric space $X$ is a metric space if and only if every sequence in $X$ has at most one limit.

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

  $$\forall ϵ>0\exists \delta >0\forall x^{\prime }\in X:(d_X(x,x^{\prime })\le \delta ⟹d_Y(f(x),f(x^{\prime }))<ϵ)$$

• continuous if it is continuous at every point of $X$, that is

  $$\forall x\in X\forall ϵ>0\exists \delta >0\forall x^{\prime }\in X:(d_X(x,x^{\prime })\le \delta ⟹d_Y(f(x),f(x^{\prime }))<ϵ)$$

• uniformly continuous if

  $$\forall ϵ>0\exists \delta >0\forall x,x^{\prime }\in X:(d_X(x,x^{\prime })\le \delta ⟹d_Y(f(x),f(x^{\prime }))<ϵ)$$

• Lipschitz continuous if

  $$\exists L\ge 0\forall x,x^{\prime }\in X:d_Y(f(x),f(x^{\prime }))\le L{d}_X(x,x^{\prime })$$

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

$$B(c,r)=\{x\in X:d(x,c)<r\}.$$

The closed ball with center $c\in X$ and radius $r>0$ is the set

$$\overline{B}(c,r)=\{x\in X:d(x,c)\le r\}.$$

The sphere with center $c\in X$ and radius $r>0$ is the set

$$S(c,r)=\{x\in X:d(x,c)=r\}.$$

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 $ϵ>0$ such that $B(x,ϵ)\subset O$.

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 $d$.

• 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.

Complete Metric Spaces

• Definition
• Banach Fixed-Point Theorem
• Baire
• Metric Completion