Chapter 3: Basic Topology of R - Tarleton State University · Chapter 3: Basic Topology of R PWhite...

Chapter 3: BasicTopology of R

PWhite

Discussion

Open and ClosedSets

Compact Sets

Chapter 3: Basic Topology of R

Peter W. [email protected]

Initial development byKeith E. Emmert

Department of MathematicsTarleton State University

Spring 2021 / Real Anaylsis I


PWhite

Discussion

Open and ClosedSets

Compact Sets

Overview

Discussion: The Cantor Set

Open and Closed Sets

Compact Sets


PWhite

Discussion

Open and ClosedSets

Compact Sets

Cantor’s Monster

Definition 1I Begin with C0 = [0,1].I Let C1 = C0\

(13 ,

23

).

I Let C2 = C1\[(1

9 ,29

)∪(7

9 ,89

)].

I Let Cn be the set obtained by removing all the“middle thirds” from Cn−1, for n ≥ 1.

I Then, C =∞⋂

n=0

Cn.

The resulting set C is called the Cantor set.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Cantor’s Monster0 1

13

23

19

29

79

89

C0

C1

C2

C3

127

227

727

827

1927

1927

2527

2627

......

......

......

......

...

I C 6= ∅ since 1 ∈ Cn for all n. In fact, if x is anendpoint of Cn, then x ∈ C.

I We have removed intervals of length13 ,2 ·

19 ,4 ·

127 , . . .:

13

+ 2 · 19

+ 4 · 127

+ · · ·+ 2n−1 · 13n + · · · = 1.

Thus, the Cantor set has zero length.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Fractional DimensionI The dimension of a point is zero.I The dimension of a line segment is one.I The dimension of a square is two.I The dimension of a cube is three.I Scaling: If we magnify the size of the above

creatures by three, we obtainI One = 30 copies of the point.I Three = 31 copies of the line segment.I Nine = 32 copies of the square.I Twenty-seven 33 copies of the cube.

I In the above cases, dimension is an exponent of themagnification. So, it is reasonable to say that thenumber of copies equals the magnification raised tothe dimension.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Fractional Dimension

Now, we magnify the Cantor set by three.I Thus, we have C = [0,3], C1 = [0,1] ∪ [2,3], etc.I Notice we have two copies of the Cantor set.I Hence, if x is the dimension of the Cantor set,

magnification results in an equation

2 = 3x =⇒ x =ln(2)

ln(3)≈ 0.63093.

I So, the dimension of the Cantor set is between zeroand one.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Size of the Cantor Set

Theorem 2The Cantor set is uncountable.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Overview



Compact Sets


PWhite

Discussion

Open and ClosedSets

Compact Sets

Definitions

Remark 3Recall: Given a ∈ R and ε > 0, the ε-neighborhood of ais the set

Vε(a) = {x ∈ R | |x − a| < ε} = (a− ε,a + ε).

Definition 4A set O ⊆ R is open if for all points a ∈ O there exists anε-neighborhood Vε(a) ⊆ O.

Example 5I The sets R and ∅ are open.I The interval (a,b) is open for all a < b, a,b ∈ R. Can

you show this?I Let a,b, c,d ∈ R be such that a < b and c < d . Then

(a,b) ∪ (c,d) is open. Can you show this?


PWhite

Discussion

Open and ClosedSets

Compact Sets

Theory

Theorem 61. The union of an arbitrary collection of open sets is

open.2. The intersection of a finite collection of open sets is

open.

Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Definition

Definition 7A point x is a limit point of a set A if everyε-neighborhood Vε(x) of x intersects the set A in somepoint other than x .

Remark 8We can reformulate this definition by: x is a limit point ofa set A if for every ε > 0, (A\{x}) ∩ Vε(x) 6= ∅.

Theorem 9A point x is a limit point of a set A if and only ifx = lim

n→∞an for some sequence (an) contained in A

satisfying an 6= x for all n ∈ N.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Definitions

Definition 10A point a ∈ A is an isolated point of A if it is not a limitpoint of A.

Definition 11A set F ⊆ R is closed if it contains its limit points.

Theorem 12A set F ⊆ R is closed if and only if every Cauchysequence contained in F has a limit that is also anelement of F .Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Examples

Example 13

I Investigate the set A =

{1n| n ∈ N

}. Is it open?

Does it contain isolated points? Is it closed?I Let a,b ∈ R be such that a < b. Then [a,b] is closed.I What are the limit points of Q?I What are the limit points of the irrational numbers?I What are the limit points of R?I Is R closed? Is ∅ closed?I Is (0,5] open? Closed?I Is (0,∞) open? Closed? What about [0,∞)?

Theorem 14 (Density of Q in R)Given any y ∈ R, there exists a sequence of rationalnumbers that converges to y.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Definition

Definition 15Given a set A ⊆ R, let L be the set of all limit points of A.The closure of A is defined to be A = A ∪ L.I Q = RI (0,1) = [0,1]

Theorem 16For any A ⊆ R, the closure A is a closed set and is thesmallest closed set containing A.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

More Theory

Remark 17Recall: The complement of a set A ⊆ R is

Ac = {x ∈ R | x 6∈ A}.

Theorem 18A set O is open if and only if Oc is closed. Likewise, a setF is closed if and only if Fc is open.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

More Theory

Theorem 191. The union of a finite collection of closed sets is

closed.2. The intersection of an arbitrary collection of closed

sets is closed.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Homework

Pages: 93 – 94Problems: 3.2.2, 3.2.3, 3.2.4, 3.2.6, 3.2.8


PWhite

Discussion

Open and ClosedSets

Compact Sets

Overview



Compact Sets


PWhite

Discussion

Open and ClosedSets

Compact Sets

Definition

Definition 20A set K ⊆ R is compact if every sequence in K has asubsequence that converges to a limit that is also in K .

Example 21The interval [0,5] is compact (Use Bolzano WeierstrassTheorem)

Theorem 22 (Heine-Borel Theorem)A set K ⊆ R is compact if and only if it is closed andbounded.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Theory

Theorem 23If K1 ⊇ K2 ⊇ · · · ⊇ Kn ⊇ · · · is a nested sequence of

nonempty compact sets, then the intersection∞⋂

n=1

Kn is

not empty.Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Open Covers

Definition 24Let A ⊆ R.I An open cover for A is a collection of open sets{Oλ | λ ∈ Λ} such that A ⊆

⋃λ∈Λ

Oλ.

I Given an open cover for A, a finite subcover is afinite sub-collection of open sets from the originalopen cover whose union still manages to completelycontain A.

Example 25I Let A = {(n − 1,n + 1) | n ∈ Z}. Then A is an open

cover for R that does not have a finite subcover.I Let F = {0} ∪

{1n | n ∈ N

}. Every open cover of F

has a finite subcover.I Every open cover of [0,1] has a finite subcover.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Theory

Theorem 26Let K be a subset of R. The following are equivalent:

1. K is compact.2. K is closed and bounded.3. Any open cover of K has a finite subcover.

Proof:


PWhite

Discussion

Open and ClosedSets

Compact Sets

Example

Theorem 27Let f : R→ R be a function. The following are equivalent.

1. f is continuous if for each fixed x0, and for all ε > 0,there exists a δ > 0 such that |x − x0| < δ implies|f (x)− f (x0)| < ε.

2. if for every open set O, then f−1(O) is open.

Example 28Suppose that the continuous function f : A→ B. Supposefurther that A is compact. Prove that the image of f iscompact.


PWhite

Discussion

Open and ClosedSets

Compact Sets

Homework

Pages: 99 – 100Problems: 3.3.1, 3.3.2, 3.3.4, 3.3.6, 3.3.7

Chapter 3: Basic Topology of R - Tarleton State University · Chapter 3: Basic Topology of R PWhite...

Documents

Transcript of Chapter 3: Basic Topology of R - Tarleton State University · Chapter 3: Basic Topology of R PWhite...