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
Chapter 3: BasicTopology of R
PWhite
Discussion
Open and ClosedSets
Compact Sets
Overview
Discussion: The Cantor Set
Open and Closed Sets
Compact Sets
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
PWhite
Discussion
Open and ClosedSets
Compact Sets
Size of the Cantor Set
Theorem 2The Cantor set is uncountable.Proof:
Chapter 3: BasicTopology of R
PWhite
Discussion
Open and ClosedSets
Compact Sets
Overview
Discussion: The Cantor Set
Open and Closed Sets
Compact Sets
Chapter 3: BasicTopology of R
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?
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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
Chapter 3: BasicTopology of R
PWhite
Discussion
Open and ClosedSets
Compact Sets
Overview
Discussion: The Cantor Set
Open and Closed Sets
Compact Sets
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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:
Chapter 3: BasicTopology of R
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.
Chapter 3: BasicTopology of R
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