Elementary Analysis. Homework 4

MATH5301 Elementary Analysis. Homework 4

4.1
Let (S1, d1) and (S2, d2) be two metric spaces. Show that each of the following determines the metric on S1 ×S2
(here xj ∈ S1, yj ∈ S2):
(a) d((x1, y1),(x2, y2)) = max{d1(x1, x2), d2(y1, y2)}
(b) d((x1, y1),(x2, y2)) = d1(x1, x2) + d2(y1, y2)
(c) d((x1, y1),(x2, y2)) = p
(d1(x1, x2))2 + (d2(y1, y2))2
4.2
(a) A set A in the metric space (S, d) is called bounded, if ∃R > 0 ∧ ∃x ∈ S : A ⊂ BR(x). Prove that if A is
unbounded then there exists a sequence {xn} ⊂ A such that ∀m, n ∈ N ⇒ d(xn, xm) > 1.
(b) Show that in the normed space (V, | · |) the open unit ball Br = {x ∈ V : |x| < 1} is a convex set, i.e.
∀x, y ∈ Br, ∀t ∈ [0, 1] ⇒ tx + (1 − t)y ∈ Br.
4.3
For R
2
equipped with the usual Euclidean metric
(a) Show that D = {(x, y) : x
2 + y
2 6 1} is a closed set.
(b) Find the infinite collection of open sets {An} such that T
n
An = B1(0)
4.4
Let SR
2
. Which of those sets are open or close
A = {(x, y) : x
2 + y
2 < 1}
B = {(x, y) : x = 0 ∧ −1 6 y 6 1}
C = {(x, y) : 1 < x < 2 ∧ −1 6 y 6 1}
D = {(x, y) : |x| + |y| < 2}
E = {(x, y) : x
2 − y
2 < 1 ∧ |x| + |y| < 4}
(a) In the Euclidean metric
(b) in the Manhattan metric
(c) In the highway metric: dh((x1, y1),(x2, y2)) = (
|y1 − y2|, if x1 = x2
|y1| + |y2| + |x1 − x2|, if x1 6= x2
4.5
Let (S, d) be a metric space.
(a) Show that for all A ⊂ B ⊂ S one has int(A) ⊆ int(B), A¯ ⊆ B¯. Provide an example, showing that these
relations cannot be made strict.
(b) Is the following true: int(A ∪ B) = int(A) ∪ int(B)?
(c) Is the following true: A ∩ B = A¯ ∩ B¯?
4.6
Give a topological proof of the infinitude of the set of prime numbers. (H. Furstenberg, 1955).
Denote Na,b := {a + nb | b ∈ Z} ⊂ Z. Define the topology on Z as follows: The set U will be called open if
for any a ∈ U there exists b ∈ Z such that Na,b ⊂ U. Note that every open set is infinite.
(a) Show that it is indeed a topology, i.e. any union of open sets is open and any finite intersection of open
sets is open.
(b) Show that Na,b is closed.
(c) Show that Z \ {−1, 1} is open
(d) Prove that the set P of prime numbers cannot be finite. Hint: Z \ {−1, 1} =
S
p∈P
N0,p.