26.The Metric Space Review Sheet

17.1 Definition:

For the moment we will focus on the plane and some of its subsets. Abstracting the notion of Euclidian distance,

A Metric Space is a set $\QTR{Large}{M}$ and a function . We write

such that for all in $\QTR{Large}{M}$ :

Positivity: 0 unless in which case 0 $\QTR{Large}{.}$
Symmetry: For all
Triangle Inequality: For all

17.1.4 Lemma:

The the Triangle Inequality has a second, equivalent form:

For all

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18.1 Definition:

Fix a Metric Space $\QTR{Large}{(M,d)}$ and a point

Given 0 we define the open ball of radius around $\QTR{Large}{x,}$
We say is open if for any there exist an 0 such that
The set of open sets is called the Topology defined by the Metric.

18.2 Theorem:

In a Metric Space $\QTR{Large}{(M,d)}$

$\QTR{Large}{M}$ and (the empty set) is open
Let be an arbitrary set of open sets, then is also open.
Let be a finite set of open sets, then is also open.

18.3 Definition:

Given Metric Spaces , and a function . We say that $\QTR{Large}{f}$ is continuous a point $\QTR{Large}{x\in }$ if given any 0 there is a 0 such that

We say that $\QTR{Large}{f}$ is continuous if it is continuous at every point.

18.3.1 Theorem:

Given Metric Spaces , ,a function is continuous if and only for every open set

is open in

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19.1 Definition:

Let be a sequence in a Metric Space $\QTR{Large}{(M,d)}$ . We say the sequence converges to $\QTR{Large}{x\in }$ $\QTR{Large}{M}$ , written , if for any 0 there exists an n such that for n.

19.2 Theorem:

Given Metric Spaces $\QTR{Large}{(M,d)}$ , and a function , $\QTR{Large}{f}$ is continuous a point $\QTR{Large}{x\in }$ $\QTR{Large}{M}$ if and only if for every ,

Note that this implies that $\QTR{Large}{f}$ is is continuous at every point if and only if for every convergent sequence

19.3 Definition:

Given a Metric Space $\QTR{Large}{(M,d)}$ , and a subset Define

and

is called the closure of $\QTR{Large}{A}$ .

19.4 Lemma:

$\$ For any $\QTR{Large}{A}$ , is open.
That is, the closure of the closure of $\QTR{Large}{A}$ is the closure of $\QTR{Large}{A}$ .

19.5 Definition:

We call a subset dense if .

Example:

is dense in .

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20.1 Definition:

A sequence in a Metric Space $\QTR{Large}{(M,d)}$ is called a Cauchy Sequence

if for any 0 there exists an n such that if n then

20.2 Lemma:

In the setting of 20.1 , every Cauchy sequence is bounded. In particular there is a number $\QTR{Large}{b>}$ 0 and an such that

20.5 Definition:

A Metric Space is called complete if every Cauchy sequence converges. In particular,

is a complete metric space.

20.8 Definition:

Given Metric Spaces $\QTR{Large}{(M,d)}$ and and a map we say that $\QTR{Large}{f}$ is

uniformly continuous if for all $\QTR{Large}{x\in }$ $\QTR{Large}{M}$ ,given any 0 there is a 0 such thatNote that this differs for continuity in that is independent of $\QTR{Large}{x}$ .

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21.1 Definition:

Given a Metric Space $\QTR{Large}{(M,d)}$ , and a subset we say $\QTR{Large}{x}$ is a limit point of $\QTR{Large}{A\ }$ if

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That is $\QTR{Large}{x\ }$ is in the closure of

Note: It is not necessarily the case that the set of limit points of $\QTR{Large}{A\ }$ is the closure of $\QTR{Large}{A}$ .

For example, a singleton set $\QTR{Large}{\{x\}}$ has no limit points but is its own closure.

21.2 Definition:

A Metric Space, $\QTR{Large}{(M,d)}$ , is called compact if every infinite subset has a limit point.

21.4 Theorem:

Let $\QTR{Large}{(M,d)}$ be a Metric Space. The following are equivalent:

$\QTR{Large}{(M,d)}$ is compact.
Every sequence in $\QTR{Large}{M}$ has a convergent subsequence.

21.4.1 Corollary:

Every compact Metric Space is complete:

21.5 Theorem:

Given a compact Metric Spaces $\QTR{Large}{(M,d)}$ :

For any 0 and some n there exists points

such that .
In particular, M is bounded in the sense that there exists some b0 such that b for any .

21.6 Theorem:

Every compact metric space has a countable dense subset.

21.7 Theorem:

Given Metric Spaces $\QTR{Large}{(M,d)}$ and with $\QTR{Large}{M}$ compact, and a continuous map then $\QTR{Large}{f(M)}$ is compact.

21.8 Theorem:

Given Metric Spaces $\QTR{Large}{(M,d)}$ and with $\QTR{Large}{M}$ compact, and a continuous map then $\QTR{Large}{f}$ is also uniformly continuous.

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22.1 Definitions, Remarks, Observations, and Notation:

We will use the notation as short hand for . We will also use the notation for indexed sets of sequences
Given a Metric Space $\QTR{Large}{(M,d)}$ and subsets , we define

and

In general takes values in
Given sequences ₁ $\$ and it will be convenient to define
Finally we define

Note that if is finite for any n ,then is finite since

0
Note that, almost by definition, a sequence is a Cauchy sequence if and only if 0.
Given a Metric Space $\QTR{Large}{(M,d)}$ we let denote its Set of Cauchy Sequences.

22.2 Theorem:

is well defined on the Set ,and satisfies the following metric properties :

22.3 Definitions, Observations, and Notation:

Based on 22.2, we can define an equivalence relation on by setting 0. We use the notation for .
One checks that induces a metric on . We use the same notation for this induced metric. On the other hend, we will use to denote the equivalence class of Cauchy sequences containing .
Define a Set map by the formula

, the equivalence class of ,where for all

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23.3 Definition:

A Metric Space, $\QTR{Large}{(M,d)}$ is called

limit compact if every infinite subset has a limit point.
sequentially compact if Every sequence in $\QTR{Large}{M}$ has a convergent subsequence. That is given we can find a subsequence
countably compact if every covering by a countable number of open sets , , contains a finite subcover. That is there is some finite subset such that
"totally"(my term) compact if every covering by a open sets contains a finite subcover.

23.4 Theorem:

For Metric Spaces these four forms of compactness are equivalent.

In General Topology these four forms of compactness are not equivalent.

23.5 Theorem: (Lebesgue Number)

Given a sequentially/limit compact Metric Space $\QTR{Large}{(M,d)}$ and a covering by open sets

( ), there exists a real number such that every open ball of radius is contained in some element of .

The number is called a Lebesgue number for the covering.

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24.1 Definition:

Given $\QTR{Large}{(M,d)}$ a Metric Space, a function is said to be a contraction mapping if there is a constant $\QTR{Large}{q}$ with such that for all

24. 2 Theorem:(Banach Fixed Point Theorem)

Let $\QTR{Large}{(M,d)}$ be a complete metric space then every contraction has a unique fixed point.

Proof:

Thus is Cauchy. Moreover we can use * to estimate the limit.

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25.1 Definition:

Let be a sequence of continuous functions for a metric space $\QTR{Large}{(M,d)}$ to a metric space . The sequence is said to converge uniformly to if for any 0 , there is a n such that for all and i n .