Chapter 3: Basic Topology of R

Theorem 3.2.5: What property do all limit point have?

A point x is a limit point of a set A iff x is the limit of sequence contained in A such that x is not an element of that sequence (pg. 90)

Definition 3.2.7: What is a closed set?

A set A is closed if every limit point of A lives in A (pg. 90)

Definition 3.4.4: What is a disconnected set?

A set E is disconnected if it can be written as the union of two separated sets (pg. 104)

Definition 3.3.1: What is a compact set?

A set K is compact if every sequence in K has a subsequence that converges to a limit that is also in K(pg. 96)

Theorem 3.3.4: (The Characterization of compactness in R) A set K is compact iff ...

A set K is compact iff K is closed and bounded. The forward direction is proved in the chapter while the converse is proven in exercise 3.3.3 (pg. 96)

Theorem 3.3.8: (Heine-Borel Theorem)

A set K is compact iff it is bounded and closed iff every open cover for K has a finite subcover (pg. 98)

Theorem 3.4.6: A set E ⊆ R is connected iff for all nonempty disjoint sets A and B...

See exercise 3.4.6 for proof (pg. 104)

Definition 3.2.4: What is a limit point?

Notice that limit points of a set DO NOT have to live in the set (pg. 90)

Definition 3.3.6 What is an open cover for a set A?

(pg. 98)

Classify the following sets as open, closed, or neither (i) the set of reals (ii) the set of rationals (iii) the set of naturals (iv) the set of integers

(i) The set of reals is open and closed. It's open because given any real number, there exists an epsilon neighborhood that is contained in the reals. It's closed because the reals contains all of its limit points (ii) The rationals are neither open nor closed. Given an epsilon neighborhood around any rational number, there exists an irrational number in that neighborhood; this implies the rationals are not open. The rationals are not closed because the any irrational number a is a limit point of the rationals, but a does not live in the irrationals. (iii) The set of naturals is closed vacuously (iv) The set of integers is closed vacuously

Theorem 3.2.3: What can we say about the union and of an arbitrary collection of open sets? What can we say about a the intersection of a finite number of open sets?

(i) The union of an arbitrary collection of open sets is open (ii) The intersection of a finite collection of open sets is open This is easily proven using the definitions of union/intersection and the definition of an open set. (pg. 89)

Definition 3.4.4: What does it mean for two sets A, B to be separated? What is a disconnected set? What is a connected set?

(pg. 104)

Definition 3.4.4: what is a connected set?

(pg. 104)

Theorem 3.4.7: A set E⊆R is connected iff whenever a < c < b with a,b∈E, ...

(pg. 105)

What is an epsilon neighborhood?

(pg. 88)

Example 3.2.9(ii): prove that the closed interval [c, d] is a closed set.

(pg. 90)

Theorem 3.2.8:

(pg. 90)

Definition 3.2.11: What is the closure of a set A?

(pg. 91)

Theorem 3.2.10: The density of Q in R

(pg. 91)

Theorem 3.2.12

(pg. 92)

Theorem 3.2.13: What can we say about the complement of a closed set? What can we say about the complement of an open set?

(pg. 92)

Redo problem 3.2.2

(pg. 93)

Redo problem 3.2.6

(pg. 94)

Redo problem 3.2.8

(pg. 94)

Definition 3.3.3: What is a bounded set?

(pg. 96)

Exercise 3.3.2: Which of the following are compact sets? 1) The set of natural numbers 2) The rationals intersected with [0, 1]

1) The set of natural is not compact since they are unbounded. 2) The set of rationals between 0 and 1 (including 0 and 1) is not compact because it does not contain all of its limit points, namely the irrational numbers between 0 and 1

Give an example of two separated sets

A = (0,1) B = (1, 2) (pg. 104)

Definition 3.4.1: What is a perfect set? Give an example

A set is perfect if it is closed and contains no isolated points. Any closed interval (other than singleton sets) are perfect sets (pg. 102)

Definition 3.2.1: What is an open set?

A subset O of the real numbers is open if for every element a of O there exists an epsilon neighborhood centered at a that lives in O (pg. 88)

Definition 3.2.6: What is an isolated point?

An point x is an isolated point of a set A if it is not a limit point of A. Isolated points MUST live in A (pg. 90)

Example 3.3.2: Give an example of a compact set

Any closed interval (pg. 96)

Explain why any nonempty finite perfect set can't have a finite number of elements.

Any nonempty finite set contains isolated points and is thus not perfect

Theorem 3.4.3: What can we say about the cardinality of any nonempty perfect set?

Any nonempty perfect set P is uncountable. To prove this, assume that P is countable and derive a contradiction (pg. 102)

Theorem 3.3.5: (Nested Compact Set Property)

Compare to Nested Interval Property (pg. 97)

Example 3.2.9(i): Let A = {1/n}. Describe the isolated points and limit points of A. Is A closed?

Each element of A is an isolated point. The only limit point of A is x=0 and it does not live in A. Thus A is closed (pg. 90)

Which of the following are open sets? (i) (0, 1) (ii) (0, 1] (iii) [0, 1] (iv) (0, 1) U (0, 1] (v) (0, 1) U {2}

FIXME (pg. 88)

T/F: If a set is not open, it must be closed.

False, a set doesn't have to be open or closed. E.g. the set A = {1/n: n = 1, 2, 3, ...} is not open because there none of the epsilon neighborhoods for any 1/n are subsets of A. Likewise, A is not closed because 0 is a terminal point of A but 0∉A

Is the half-open interval A = (c, d] open or closed?

It is neither open nor closed. It's not open since any epsilon neighborhood of d is not a subset of A. It's not closed since c is a limit point of A but c is not in A (pg. 92)

Is the set of real numbers open or closed?

Its both open and closed

Does a limit point of a set have to live in the set?

Nope. e.g. the point x=1 is a limit point of the set A = (0,1)U(1,2)

Exercise 3.4.1: Let P be perfect and K be compact. Is the intersection P∩K always compact? Always perfect?

Since P is perfect, P is compact. Therefore, the intersection P∩K is an intersection of two compact sets which is always compact.

Fill in the blank: The arbitrary (blank) of open sets is always open. The arbitrary (blank) of closed sets is always closed.

The arbitrary UNION of open sets is always open. The arbitrary INTERSECTION of closed sets is always closed. See theorem 3.2.3 and 3.2.14

Give an example of a set that is not compact

The interval [0, oo) (class)

Which two sets are both open and closed?

The real numbers and the empty set (online)

What is the set of limit points of the rational numbers? What is the closure of the rational numbers?

The real numbers. The closure is also the real number. This is because the rational numbers are dense in the reals.

Example: What are the limit points of the set A = (0, 1]∩{2} ?

The set of points (0, 1]. 2 is not a limit point (thus it's an isolated point) of A since the epsilon neighborhood {real x: |2-x|<1/2} does not intersect the set A at some point other than 2. (class)

Example 3.2.9(iii): What is the set of limit points of the set of rational numbers?

The set of real numbers. We demonstrate this using the fact that the rationals are dense in the reals (pg. 90)

Theorem 3.2.14: What can we say about the union of a collection of closed sets? What about the intersection of a collection of closed sets?

The union of a finite collection of closed sets is closed. The intersection of an arbitrary collection of closed sets is closed (pg. 93)

T/F: if a set is not open, it must be closed

This is false. For example the set A = (0, 1] is neither open nor closed. A is not open because there isn't an epsilon neighborhood around 1 that is completely in A. A is not closed because 0 is a limit point of A but 0 is not in A

T/F: The intersection A∩B of two compact sets A and B is always compact

True. Since A and B are both compact, they are both bounded and closed. Therefore, their intersection is both bounded and closed and is therefore compact

T/F: The set of rationals is disconnected

True: see example 3.4.5(ii) (pg. 104)

Give examples of open sets

Which of the following are open? (i) The set of real numbers (ii) The empty set (iii) The interval (a, b), where a < b (pg. 88)

Is the empty set perfect?

Yes, because it is closed and has no isolated points

Example 3.4.2: Is the cantor set perfect?

Yes, we can show that the cantor set is closed and contains no isolated points (pg. 102)