Lecture 10: Topology of the Real Line

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Lecture 10: Top ology of the Real Line

10.1 Intervals

De nition Given anytwo extended real numb ers a

a; b=fx : x 2 R;a

is called an open interval. Given anytwo nite real numb ers a b, the sets

[a; b]=fx : x 2 R;a x bg;

1;b]=fx : x 2 R;x bg;

and

[a; +1=fx : x 2 R;x ag

are called closed intervals.Any set whichisanopeninterval, a closed interval, or is given

by, for some nite real numb ers a

a; b]=fx : x 2 R;a

[a; b=fx : x 2 R;a x

is called an interval.

Prop osition If a; b 2 R, then

a; b=fx : x = a +1 b; 0 < < 1g:

Pro of Supp ose x = a +1 b for some 0 <<1. Then

b x = b a = b a > 0;

so x

x a = 1a 1 b =1 b a > 0;

so a

Now supp ose x 2 a; b. Then

b x x a b x b x

x = a + b = a + 1 b

b a b a b a b a

and

b x

0 < < 1:

b a 10-1

Lecture 10: Top ology of the Real Line 10-2

10.2 Op en sets

De nition A set U R is said to b e open if for every x 2 U there exists >0 such

that

x ; x + U:

Prop osition Every op en interval I is an op en set.

Pro of Supp ose I =a; b where a

b e the smaller of x a and b x. Supp ose y 2 x ; x + . If b =+1, then b>y;

otherwise, wehave

b y>b x + =b x b x b x=0;

so b>y.Ifa = 1, then a

y a>x a =x a x a x a=0;

so a

Note that R is an op en set it is, in fact, the op en interval 1; +1, as is ; it

satis es the de nition trivially.

Prop osition Supp ose A is a set and, for each 2 A, U is an op en set. Then U

is an op en set.

U . Then x 2 U for some 2 A. Since U is op en, there exists Pro of Let x 2

an >0 such that x ; x + U .Thus

[

x ; x + U U :

Hence U is op en.

Prop osition Supp ose U ;U ;::: ;U is a nite collection of op en sets. Then U is

1 2 n i

i=1

op en.

Pro of Let x 2 U . Then x 2 U for every i =1; 2;:::;n.For each i,cho ose > 0

i i i

i=1

such that x ;x + U . Let b e the smallest of ; ;:::; . Then >0 and

i i i 1 2 n

x ; x + x ;x + U

i i i

for every i =1; 2;::: ;n.Thus

x ; x + U :

i=1

Hence U is an op en set.

i=1

Lecture 10: Top ology of the Real Line 10-3

Along with the facts that R and ; are b oth op en sets, the last two prop ositions show

that the collection of op en subsets of R form a topology.

De nition Let A R.Wesay x 2 A is an interior p ointof A if there exists an >0

such that x ; x + A. The set of all interior p oints of A is called the interior of A,

denoted A .

Exercise 10.2.1

Show that if A R, then A is op en.

Exercise 10.2.2

Show that A is op en if and only if A = A .

Exercise 10.2.3

Let U R b e a nonempty op en set. Show that sup U=2 U and inf U=2 U .

10.3 Closed sets

De nition A p oint x 2 R is called a limit point of a set A R if for every >0 there

exists a 2 A, a 6= x, such that a 2 x ; x + .

De nition Supp ose A R. A p oint a 2 A is called an isolatedpoint of A if there exists

an >0 such that

A \ x ; x + =fag:

Exercise 10.3.1

Identify the limit p oints and isolated p oints of the following sets:

a [1; 1],

b 1; 1,

 

c : n 2 Z ,

d Z,

e Q.

Exercise 10.3.2

Supp ose x is a limit p oint of the set A. Show that for every >0, the set x ; x + \ A

is in nite.

We denote the set of limit p oints of a set A by A .

De nition Given a set A R, the set A = A [ A is called the closure of A.

De nition A set C R is said to b e closed if C = C .

Prop osition If A R, then A is closed.

Pro of Supp ose x is a limit p ointof A.We will show that x is a limit p ointof A, and

hence x 2 A.Now for any >0, there exists a 2 A, a 6= x, such that

 

;x + : a 2 x

2 2

Lecture 10: Top ology of the Real Line 10-4

If a=2 A, then a is a limit p ointof A, so there exists b 2 A, b 6= a and b 6= x, such that

 

b 2 a ;a + :

2 2

Then

 

jx bjjx aj + ja bj < + = :

2 2

A is closed. Hence x 2 A , and so

Prop osition A set C R is closed if and only if for every convergent sequence fa g

k k 2K

with a 2 C for all k 2 K ,

lim a 2 C:

k !1

Pro of Supp ose C is closed and fa g is a convergent sequence with a 2 C for all

k k 2K k

k 2 K . Let x = lim a .Ifx = a for some integer k , then x 2 C . Otherwise, for

k !1 k k

every >0, there exists an integer N such that ja xj <. Hence a 6= x and

N N

a 2 x ; x + :

Thus x is a limit p ointof C , and so x 2 C since C is closed.

Now supp ose that for every convergent sequence fa g with a 2 C for all k 2 K ,

k k 2K k

lim a 2 C: Let x b e a limit p ointofC .For k =1; 2; 3;:::,cho ose a 2 C such that

k !1 k k

1 1

a 2 x ;x + . Then clearly

k k

x = lim a ;

k !1

so x 2 C .Thus C is closed.

Exercise 10.3.3

Show that every closed interval I is a closed set.

C Prop osition Supp ose A is a set and, for each 2 A, C is a closed set. Then

is a closed set.

Pro of Supp ose x is a limit p ointof C . Then for any >0, there exists y 2

C such that y 6= x and y 2 x ; x + . But then for any 2 A, y 2 C ,so x

is a limit p ointofC . Since C is closed, it follows that x 2 C for every 2 A.Thus

T T

x 2 C and C is closed.

2A 2A

C Prop osition Supp ose C ;C ;:::;C is a nite collection of closed sets. Then

i 1 2 n

i=1

is closed.

Pro of Supp ose fa g is a convergent sequence with a 2 C for every k 2 K .

k k 2K k i

i=1

Let L = lim a . Since K is an in nite set, there must exist an integer m and a

k !1 k

subsequence fa g such that a 2 C for j =1; 2;:::. Since every subsequence of

n n m

j j

j =1

fa g converges to L, fa must converge to L. Since C is closed, g

k k 2K n m

j =1

[

L = lim a 2 C C :

n m i

j !1

i=1

Lecture 10: Top ology of the Real Line 10-5

C is closed. Thus

i=1

Note that b oth R and ; satisfy the de nition of a closed set.

Prop osition A set C R is closed if and only if R n C is op en.

Pro of Assume C is closed and let U = R n C .IfC = R, then U = ;, which is op en; if

C = ;, then U = R, which is op en. So wemay assume b oth C and U are nonempty. Let

x 2 U . Then x is not a limit p ointof C , so there exists an >0 such that

x ; x + \ C = ;:

Thus

x ; x + U;

so U is op en.

Now supp ose U = R n C is op en. If U = R, then C = ;, which is closed; if U = ;, then

C = R, which is closed. So wemay assume b oth U and C are nonempty. Let x b e a limit

p ointof C . Then, for every >0,

x ; x + \ C 6= ;:

Hence there do es not exist >0 such that

x ; x + U:

Thus x=2 U ,so x 2 C and C is closed.

Exercise 10.3.4

1 n+1

For n =1; 2; 3;:::, let I = ; . Is I op en or closed?

n n

n=1

n n

Exercise 10.3.5

n1 1

I op en or closed? ; ]. Is For n =3; 4; 5;:::, let I =[

n n

n=3

n n

Exercise 10.3.6

Supp ose, for n =1; 2; 3;:::, the intervals I =[a ;b ] are such that I I . If

n n n n+1 n

+ +

a = supfa : n 2 Z g and b = inf fb : n 2 Z g, show that

n n

I =[a; b]:

n=1

Exercise 10.3.7

Find a sequence I , n =1; 2; 3;:::, of closed intervals such that I I for n =1; 2; 3;:::

n n+1 n

and I = ;.

n=1

Exercise 10.3.8

Find a sequence I , n =1; 2; 3;:::, of b ounded, op en intervals such that I I for

n n+1 n

n =1; 2; 3;::: and I = ;.

n=1

Lecture 10: Top ology of the Real Line 10-6

Exercise 10.3.9

S S

n n

Supp ose A R, i =1; 2;:::;n, and let B = B = A . A . Show that

i i i

i=1 i=1

Exercise 10.3.10

S S

1 1

Supp ose A R, i 2 Z , and let B = A . Show that A B . Find an example

i i i

i=1 i=1

for which B 6= A .

i=1

Exercise 10.3.11

Supp ose U R is a nonempty op en set.For each x 2 U , let

[

J = x ; x + ;

where the union is taken over all >0 and >0 such that x ; x + U .

a Show that for every x; y 2 U , either J \ J = ;,orJ = J .

x y x y

J , where B U is either nite or countable. b Show that U =

x2B