# 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:
1. $d(x,y)=0 \iff x=y$,
2. $d(x,y)=d(y,x)$,
3. $d(x,z)\leq d(x,y) + d(y, z)$ (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
1. $x\pm y=(x_1\pm y_1, x_2\pm y_2, \ldots, x_d\pm y_d)$,
2. $|x|=\sqrt{x_1^2+x_2^2+\cdots+x_d^2}$.
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:
1. Any isometry with an invariant point is a rotation or a reflection.
2. 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:
1. A contraction mapping $f$ on a complete nonempty metric space $S$ has a unique fixed point.
2. 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:
1. There is a countable set $D$ dense in $S$.
2. There is a countable base for the open sets of $S$. (second axiom of countability)
3. 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:
1. $S$ is sequentially compact,
2. $S$ is countably compact,
3. $S$ is bicompact.
A metric space $S$ is called compact iff has one (and therefore all) of the properties above.

## References

1. Measure, Topology, and Fractal Geometry, Edgar Gerald.

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