Definitions

Question 1

What axioms must a metric space satisfy?

Answer 1

(M1) d(x, y) ≥ 0 for all x, y ∈ X; d(x, y) = 0 if and only if x = y

(M2) (Symmetry) for all x, y ∈ X, d(y, x) = d(x, y)

(M3) (Triangle inequality) for all x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z)

Question 2

What is the open ball in X of radius r centred at x₀ expressed as?

Answer 2

The set B(x₀, r) = {x ∈ X: d(x,x₀) < r}

Question 3

What is the closed ball in X of radius r centre at x₀ expressed as?

Answer 3

The set B’(x₀, r) = {x ∈ X: d(x, x₀) ≤ r}

Question 4

What is the sphere in X of radius r centred at x₀ expressed as?

Answer 4

The set S(x₀, r) = {x ∈ X: d(x, x0) = r}

Question 5

When is a subset A of a metric space X bounded?

Answer 5

If A ⊆ B(x0, r) for some x₀ ∈ X and r > 0.

Question 6

When is a subset in a metric space open?

Answer 6

It is open iff it is a union of open balls.

Question 7

When is a subset V of a metric space X closed in X?

Answer 7

When X\V is open in X

Question 8

Let (X, d_X) and (Y, d_Y) be metric spaces. A function f: X → Y is an isometric embedding if, for every x₁, x₂ ∈ X:

Answer 8

d_Y(f(x₁),f(x₂)) = d_X(x₁, x₂)

Question 9

Is an isometric embedding injective?

Answer 9

Yes

Question 10

The composition of two isometries is…?

Answer 10

An isometry

Question 11

Define the notion of continuity between two metric spaces (X, d_X) and (Y,d_Y) with the function f: X → Y, a mapping between them.

Answer 11

f is continous at x₀ ∈ X if for any given ε > 0, there exists δ > 0 such that whenever d_X(x,x₀) < δ we have that d_Y(f(x),f(x₀)) < ε .

f is continuous if f is continous at every x₀ ∈ X

Question 12

When is a function f a homeomorphism?

Answer 12

f: X → Y is a homeomorphism if it is a bijection and both f and f^-1 are continuous.

Question 13

Is an isometric embedding continuous?

Answer 13

Yes

Question 14

Is every homeomorphism an isometry?

Answer 14

No

Question 15

When is a sequence (x_n) in a metric space called Cauchy?

Answer 15

If for every ε > 0 there exists N ∈ ℕ such that whenever m ≥ n ≥ N

d(x_m, x_n) < ε

Question 16

Is every convergent sequence Cauchy?

Answer 16

Yes.

Let (x_n) be a sequence in a metric space (X, d) that converges to a point a ∈ X. Let ε > 0. There eists N such that for every n ≥ N, d(x_n, a) < ε/2. It follows that for ever m ≥ n ≥ N,

d(x_n, x_m) ≤ d(x_n, a) + d(x_m, a) < ε/2 + ε/2 = ε

Question 17

When is a metric space complete?

Answer 17

If every Cauchy sequence in X converges to a point in X

Question 18

What three axioms must the non-empty set X together with a family T of subsets of X satisfy to be a toological space?

Answer 18

(T1) X, ∅ ∈ T

(T2) U, V ∈ T ⇒ U ∩ V ∈ T

(T3) U_i ∈ T for all i ∈ I ⇒∪U_i∈ T