2) Set Properties and Metric Analysis

Question 1

What is an open set

Answer 1

Question 2

Describe the proof that every open ball Br(x) is open in X

Answer 2

Question 3

Prove by verifying the definition that, for all x ∈ X, r > 0 the set A = {y ∈ X : d(y, x) > r} is open

Answer 3

Question 4

Describe the proof that any union of two open sets is open

Answer 4

Question 5

Describe the proof that the complement of a closed ball is open

Answer 5

Question 6

Explain the condition under which a subset U of a metric space (X,d) is considered open

Answer 6

U ⊆ X is open in (X, d) iff it is a union of open balls

Question 7

If (X,d1) and (X,d2) have lipchitz equivalent d1 and d2, under what condition is U ⊆ X open with respect to d1

Answer 7

If it is open with respect to d2

Question 8

What are the main properties of open sets

Answer 8

Question 9

Provide an example of an infinite intersection of open sets that is not an open set

Answer 9

• (0, 1/n) for n = 1,2,3…
• The intersection ⋂(0, 1/n) is the set {0}
• The set {0} is not open because there does not exist an interval around 0 that is completely contained within {0}

Question 10

What is an interior point and the interior

Answer 10

Question 11

What are the properties of the interior

Answer 11

Question 12

Prove the properties of the interior

Answer 12

Question 13

What is a closed set

Answer 13

Question 14

What are the main properties of closed sets

Answer 14

Question 15

What is a closure point and the closure

Answer 15

Question 16

Describe the proof that a set V is closed in X iff V = V_

Answer 16

Question 17

What are the properties of the closure

Answer 17

Question 18

Describe the proof of the properties of the closure

Answer 18

Question 19

When does a sequence converge

Answer 19

Question 20

What property does the limit of a convergent sequence have

Answer 20

It is unique

Question 21

Describe the proof that the limit of a convergent sequence is unique

Answer 21

Question 22

Suppose that Y ⊆ X and y ∈ X, then their under what condition does y lie in Y_

Answer 22

If there exists a sequence (yn)n≥1 in Y such that yn -> y as n -> ∞

Question 23

What is a Cauchy sequence in metric spaces

Answer 23

Question 24

What is a condition that implies a sequence is a cauchy sequence

Answer 24

Question 25

Describe the proof that if xn -> x in (X,d), then (xn)n≥1 is a Cauchy Sequence

Answer 25

Question 26

What does it mean for a subset to be dense

Answer 26

Question 27

What does it mean for a metric space to be bounded

Answer 27

Question 28

When is a function space bounded

Answer 28

When it’s image is bounded

Question 29

What is the diameter of a subset

Answer 29

Question 30

What is a boundary point and the boundary

Answer 30

Question 31

What are the properties of the boundary of (X,d)

Answer 31

Question 32

Prove that ∂U may be expressed as
U_ \U◦

Answer 32