A Quick Introduction to Metric Spaces

Fundamental Definitions

A metric space is a set $S $ together with a function $d: S \times S \to [0,\infty) $ satisfying three conditions:

$d(x,y)=0 \iff x=y $,
$d(x,y)=d(y,x) $,
$d(x,z)\leq d(x,y) + d(y, z) $ (Triangle inequality).

The nonnegative real number $d(x,y)$ is called the distance between $x $ and $y $. The function $d $ is called a metric on the set $S $, also a metric space can be written as a pair $(S, d)$.

The set $\mathbb{R}$ of real numbers, with the function $$ d(x,y)=|x-y| $$ is a metric space.

Let d be a positive integer. Then $\mathbb{R}^d $ is the set of all ordered $d $-tuples of real numbers. For $x=(x_1,x_2,\ldots,x_d)\in\mathbb{R}^d $ and $y=(y_1,y_2,\ldots,y_d)\in\mathbb{R}^d $, define

$x\pm y=(x_1\pm y_1, x_2\pm y_2, \ldots, x_d\pm y_d) $,
$|x|=\sqrt{x_1^2+x_2^2+\cdots+x_d^2} $.

Then we define $d-$dimensional Euclidean space to be the set $\mathbb{R}^d $ with the metric $d(x,y)=|x-y| $.

Let $x_1,x_2,\ldots,y_1,y_2,\ldots y_d $ be 2-dimensional real numbers. Then Cauchy's inequality is defined as $$ \left(\sum_{j=1}^d x_j y_j\right)^2\leq\left(\sum_{j=1}^d x_j^2\right)\left(\sum_{j=1}^d y_j^2\right) $$

Let $x,y\in\mathbb{R}^d $. Then Minkowski's inequality is defined as $$ |x+y|\leq|x|+|y|. $$

I'll give more fundamental definitions.

Let $S $ be a metric space with metric $d $, and $T\subseteq S $. Then $T $ is also a metric space with metric $$ d_T(x,y)=d(x,y), \quad x,y\in T. $$

Let $S $ be a metric space.

The diameter of $A\subseteq S$ is $$ \text{diam}A=\sup\{d(x,y): x,y\in A\}. $$
The distance between nonempty sets $A, B \subseteq S $ is $$ \text{dist}(A, B)=inf\{d(x,y):x\in A,y\in B\}. $$
Let $x\in S $ and $r>0 $. The open ball with center $x $ and radius $r $ is defined as $$ B_r(x)=\{y\in S:d(y,x)< r \}, $$ the closed ball with same center and radius is defined as $$ \overline{B_r}(x)=\{y\in S:d(y,x)\leq r \}. $$
Let $A\subseteq S$. An interior point of $A $ is a point $x $ so that $$ B_\epsilon(x)\subseteq A $$ for some $\epsilon>0$. A set $A $ is called an open set iff every point of $A$ is an interior point.
Let $A\subseteq S$ and $x\in A$. Then $x $ is a accumulation point of $A $ iff for every $\epsilon>0 $ $$A\cap(B_\epsilon(x)\setminus\{x\})\neq\emptyset.$$ $A $ is closed iff it contains all of its accumulation points.
A family $\mathcal{B}$ of open sets of a metric space $S $ is called a base for the open sets of $S $ iff, for every open set $A\subseteq S$, and every $x\in A$, there is $U\in\mathcal{B}$ such that $x\in U\subseteq A$.

Let $S$ be a metric space. Then following statements are true:

$\emptyset$ and $S $ are open sets.
If $U $ and $V $ are open sets, then $U\cap V$ is also an open set.
If $\mathcal{U} $ is any family of open sets, then $$ \bigcup_{U\in\mathcal{U}} $$ is also an open set.

Let $S$ be a metric space. Then following statements are true:

$\emptyset$ and $S $ are closed sets.
If $U $ and $V $ are closed sets, then $U\cup V$ is also a closed set.
If $\mathcal{U} $ is any family of closed sets, then $$ \bigcap_{U\in\mathcal{U}} $$ is also a closed set.

Functions on Metric Spaces

Suppose $S$ and $T $ are metric spaces. A function $h:S\to T $ is an isometry iff $$ d_T(h(x), h(y)) = d_S(x,y) $$ for all $x,y\in S $. Two metric spaces are called as isometric iff there is an isometry of one onto the other. A property is called a metric property iff it is preserved by isometry.

The following statements are true:

Any isometry with an invariant point is a rotation or a reflection.
An isometry with no invariant point is a translation or a glide-reflection.

A function $h:S\to T$ is a similarity iff there is a positive number $r$, called as ratio of $h $, such that $$ d(h(x), h(y)) = rd(x,y) $$ for all $x,y\in S $. Two metric spaces are called as similar iff there is a similarity of one onto the other.

Let $S $ and $T $ be metric spaces and $x\in S$. A function $h:S\to T$ is continous at $x$ iff, for every $\epsilon >0$, there is $\delta >0$ such that for all $y\in S$ $$ d(x,y)<\delta \Rightarrow d(h(x), h(y)) < \epsilon $$ The function $h$ is called continous iff it is continuous at every point $x\in S$.

A function $h:S\to T$ is continous iff $h^{-1}\left[V\right]$ is open in $S$ for all $V $ open in $T $.

A function $h:S\to T$ is a homeomorphism of $S$ onto $T$ iff it is bijective, and both $h$ and $h^{-1}$ are continous. Two metric spaces are called as homeomorphic iff there is an homeomorphism of one onto the other. A property is called a topological property iff it is preserved by homeomorphism.

Sequences in Metric Spaces

Let $S $ be a set. A sequence in $S $ is a function $f:\mathbb{N}\to S$ and it is defined by the infinite list of values $f(1),f(2),\cdots$ and can be showed as $$ (x_n)_{n\in\mathbb{N}}. $$ A sequence $(x_n)$ in a metric space $S $ converges to the point $x\in S$ iff for every $\epsilon>0$, there is $N\in\mathbb{N}$ such that $d(x_n,x)<\epsilon$ for all $n\geq N$, can be written as $$ \lim_{n\to\infty}x_n=x. $$ We say that the sequence is convergent iff it converges to some point.

Let $(x_n)$ be a sequence in a metric space $S$. If $x_n\to a$ and $x_n\to b$, then $a=b$.

Let $S $ and $T $ be metric spaces, and let $h:S\to T$ be a function. Then $h $ is continuous iff for every sequence $(x_n)$ in S, $$ x_n\to x \Longrightarrow h(x_n)\to h(x). $$

Let $(x_n)$ be a sequence. If we choose an infinite subset of positive integers and list them in ascending order $k_1<k_2<k_3<\cdots$, we form a new sequence, called a subsequence of $(x_n)$ $$ (x_{k_i})_{i\in\mathbb{N}}. $$

Let $x_n$ be a sequence in a metric space $S $ and $x\in S$. We say that $x $ is a cluster point of the sequence $(x_n)$ iff for every $\epsilon>0$, and every $N\in\mathbb{N}$, there exists $n\geq N$ with $d(x_n,x)<\epsilon$.

The point $x$ is a cluster point of the sequence $(x_n)$ iff $x$ is the limit of some subsequence of $(x_n)$.

Completeness

A Cauchy sequence in a metric space $S$ is a sequence $(x_n)$ satisfying; for every $\epsilon>0$ there is $N\in\mathbb{N}$ so that $d(x_n,x_m)<\epsilon$ for all $n,m\geq N$.

Every convergent sequence is a Cauchy sequence.

A metric space $S$ is complete iff every Cauchy sequence in $S$ converges in $S$.

The closure of a ser $A$ is the set $\bar{A}$, consisting of $A $ together with all of its accumulation points. A set $A$ is dense in a set $B$ iff $\bar{A}=B$. A point $x$ that belongs both to the closure of the set $A$ and to the closure of the complementary set $S\setminus A$ is called a boundary point of $A$ and can be written as $\partial A$.

Contraction Mapping

A point $x$ is a fixed point of a function $f$ iff $f(x)=x$. A function $f:S\to S$ is a contraction iff there is a constant $r<1$ such that $$ d(f(x), f(y)) \leq rd(x,y) $$ for all $x,y\in S$.

The following statements are true:

A contraction mapping $f$ on a complete nonempty metric space $S$ has a unique fixed point.
Let $f$ be a contraction mapping on a complete metric space $S$. If $x_0$ is any point of $S$, and $$ x_{n+1}=f(x_n), \quad n\geq 0, $$ then the sequence $x_n$ converges to the fixed point of $f$.

A function $f:S\to T$ is a Lipschitz function iff there is a constant $B$ with $$ d(f(x),f(y))\leq Bd(x,y), \forall x,y\in S. $$ A function $f:S\to T$ is a inverse Lipschitz function iff there is a constant $A>0$ with $$ d(f(x),f(y))\geq Ad(x,y), \forall x,y\in S. $$ A function $f:S\to T$ is a lipeomorphism iff it is both Lipschitz and inverse Lipschitz. Such a function $f$ is also called a metric equivalence.

Seperable and Compact Spaces

A family $\mathcal{U}$ of subsets of $S$ is said to cover a set iff $A$ is contained in the union of the family $\mathcal{U}$. A family which covers a set is called as a cover of the set. A cover consisting of a finite number of sets is called a finite cover. A cover consisting of a countable number of sets is called a countable cover. An open cover of a set $A $ is a cover of $A$ consisting only of open sets. If $\mathcal{U}$ is a cover of $A$, then a subcover is a subfamily of $\mathcal{U}$ that still covers $A$.

Let $S$ be a metric space. Then the following are equivalent:

There is a countable set $D$ dense in $S$.
There is a countable base for the open sets of $S$. (second axiom of countability)
Every open cover of $S$ has a countable subcover. (Lindelöf property)

A metric space $S$ is seperable iff it has one (and therefore all) of the properties above.

A metric space $S$ is called sequentially compact iff every sequence in $S$ has at least one cluster point (in $S$). A metric space $S$ is called countably compact iff every infinite subset of $S$ has at least one accumulation point (in $S$). Let $\mathcal{F}$ be a family of subsets of a set $S$. We say that $\mathcal{F}$ has the finite intersection property iff any intersection of finitely many sets from $\mathcal{F}$ is nonempty. A metric space $S $ is called bicompact iff every family of closed sets with the finite intersection property has nonempty intersection.

(Heine–Borel theorem) Let $a<b$ be real numbers. Let $\mathcal{F}$ be a family of closed subsets of of the interval $[a, b]$. If $\mathcal{F}$ has the finite intersection property, then the intersection $$ \bigcap_{F\in\mathcal{F}}F $$ of the entire family is not empty.

Let $S$ be a metric space. The following are equivalent:

$S$ is sequentially compact,
$S$ is countably compact,
$S$ is bicompact.

A metric space $S$ is called compact iff has one (and therefore all) of the properties above.

References

Measure, Topology, and Fractal Geometry, Edgar Gerald.

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