## 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).

*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} $.

*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*)

*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.

*compact*iff has one (and therefore all) of the properties above.## References

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

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