--- 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