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A metric space (S, d) consists of a set Sand a distance function, d satisfying:

1. d(x, y) 0 nonnegativity

2. d(x, y) = 0 x= y nondegeneracy

3. d(x, y) = d(y, x) symmetry

4. d(x, y) d(x, z) +d(z, y) triangel inequality

Let (M, d) be a metric space

For each fixed x M and >0, the set

D(x, ) = {y M| d(x, y)< }

is called the disk about x (also neighborhood about x)

A set A M is called open if for each x A, there exists an >0 such thatD(x, ) A

A neighborhoodof a point in Mis an open set containins that point

Facts:

The intersection of a finite number of open subsets ofM is open

The union of an arbitrary collection of open subsets ofM is open

The empty set and the whole set Mare open

The discrete metric on M is d0 such that

d0(x, y) = 0 ifx = y andd0(x, y) = 1 ifx =y

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Let (M, d) be a metric space and xk a sequence in M. xk converges to apoint inMif for every open set Ucontainingx, there is an integerNsuch that

k N = xk U

Equivilantly, a sequence xk converges to x M iff > 0, N : k N =d(x, xk)<

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The boundaryof a set A Mis defined as:

cl(A) cl(M\A)

Equivalently, the boundary ofA is the the closure ofA minus the interior ofA

Equivalently, the boundary ofA is the set of all x such that every neighborhoodofx contains at least one point in A and one point not in A, i.e. every neighh-borhood ofx contains a point in A and a point in M\A

Let Mbe a metric space. A subset A Mis called sequentially compact ifevery sequence in A has a subsequence that converges to a point in A

A cover ofA is a collection {Ui} of sets whose union contains A

A subset A of a metric space Mis called compactif every open cover ofA hasa finite subcover

Every finite set is compact, as is the empty set

A set A M is called totally bounded if for each > 0 there is a finite set{x1, x2, . . . , xN} in Msuch that the union ofD(xi, ) covers A

* Totally bounded = bounded (but not converse)

In a general metric space, M, the following are equivalent:

M is compact

Mis sequentially compact (compact sequentially compact is Bolzano-Weierstrass)

Mis complete and totally bounded

Heine-Borel Theorem: in Rn, a set is compact it is closed and bounded

Nested Set Property: Let Fk be a sequence of compact nonempty sets suchthat Fk+1 Fk for all k N. Then there is at least one point in

k=1Fk

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A map : [a, b] M, where M is a metric space, is M continuous iftk timplies (tk) (t) for every converging sequence tk in [a, b] (intuitively, a

continuous function has no breaks or jumps in its graph)

A continuous path joining two points x, y in a metric space M is a mapping: [a, b] M such that (a) = x, (b) = y and is continuous

A set in path connected if every two points in the set can be joined by acontinous path lying in the set.

Let A be a subset of a metric space M. If there exist two open setsU, V suchthat:

U V A=

A U=

A V =

A U V

Then A is said to be disconnected. If such sets do not exist, then A isconnected

Path connected = conncected

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Let (M, d) be a metric space an xk a sequence of points in M. We say xkconverge to a point x M

limk

xk= x or xkx as k

if for every open set Ucontainingx, there is an integerNsuch thatk N =xk U

Equivalantly, a sequence xk in Mconverges to x M iff

>0,N :k N = d(x, xk)<

vk v in Rn

the sequences of all the individual coordinates converge tothe correspoinding coordinate ofv in R

A set A Msi closed the limit of every sequence in A lies in A

For a set B M, x cl(B) there exists a sequence in B that convergesto x

Let (M, d) be a metric space. A Cauchy sequence is a sequence xk M suchthat

>0 N :m,n > N = d(xm, xn)<

A spaceM iscomplete every Cauchy sequence in Mconverges to a pointin M

In a normed space, a sequence xk is boundedif there is a number B such thatk, ||xk|| B

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