Reference no: EM132914596
Question 1. Let be (X; d) bounded metric space, i.e, there exist M > 0 such that d(x; y) < M for all x, y ∈ E. Let be family F ⊂ P (X) defined by:
F = {A ⊂ X, A closed and non - empty}
Let be the function:
defined by:
ρ : F × F → R
ρ(A; B) = max{supx∈B(x, A), supx∈A(x, B)}
a) Prove that ρ is a distance in F
b) If {An}n∈N sequence in (F ; ρ) convergent to A and {xn}n∈N sequence in (X; d), where xn ∈ An, convergent to x ∈ X. Show that x ∈ A
c) If {An}n∈N sequence in (F ; ρ) convergent to A. Show that:
A = ∩ ∪ Am
n≥0 m≥n
d) )Show that if X is totally bounded, then F is totally bounded.
Question 2. A space X is normal if: For all A, B closed with A∩B = ∅, there exists U, V open sets, such that A ⊂ U ,
B ⊂ V and U ∩ V = ∅.
Show that all metric space is normal. Remember the continuity of d(x, A) and d(x, B) and then construct explicity U and V to verify the request. Hint: Prove and use that for f : X → Y continue, and for all A ⊂ Y open:
f-1(A) = {x ∈ X : f (x) ∈ A} ⊂ X
is an open set too.
Question 3. Use Baire lema to prove that is not possible find a numerable family of opens {An} : n ∈ N con An ⊂ R
such that:
∩An = Q
n
Question 4. Let be (X; d) compact metric space.
a) Show that for all open cover {Ai : i ∈ I} of X, there exists s > 0 (called lebesgue number) such that:
∀x ∈ X, exists i ∈ I, B(x; s) ⊂ Ai
b) Use this result to conclude that any open coating {Ai : i ∈ I} of X has a finite subcoating.
Question 5. Let be (X; d) compact metric space.
a) Let be h : X → X an isometry, i.e, a function that for all x, y ∈ X verify that d(h(x), h(y)) =
d(x, y). Show that f (X) = X
Hint: Prove that for all x0 ∈ X, sequence xn+1 = h(xn) satisfies that, for all n m
d(xn, xm) ≥ d(x0, h(X))
b) If (Y ; ρ) is another metric space with isometries f : X → Y , g : Y → X, then f and g are bijectives.