Theorems

Question 1

Lipschitz equivalent

Answer 1

Two metrics d and e on a given set X are Lipschitz equivalent whenever there exist positive constants h, k ∈ R such that:

he(x, y) ≤ d(x, y) ≤ ke(x, y)

Question 2

A continuous function on any closed interval —->

Answer 2

Is necessarily bounded

Question 3

Every open ball Br(x) is —>

Answer 3

open in X

Question 4

Any union B1 ∪ B2 of two open balls is —>

Answer 4

open

Question 5

The compliment of a closed ball —>

Answer 5

is open

Question 6

Given any two subsets U, V ⊆ X, the following hold:

1. U ⊆ V implies —>
2. (U◦)◦ =
3. U◦ is —>
4. U◦ is —>

Answer 6

Given any two subsets U, V ⊆ X, the following hold:

1. U ⊆ V implies that U◦ ⊆ V◦
2. (U◦)◦ = U◦
3. U◦ is open in X
4. U◦ is the largest subset of U that is open in X