ASSIGNMENT 5: MATH2320/6110
DUE TIME: 5:00PM, 26 MAY 2024
Question 1. ( 14 points) Consider the ℓ2-space
ℓ2 :=
{
(xn) ∈ R∞ :
∞∑
n=1
|xn|2 <∞
}
endowed with the metric induced by the norm
∥(xn)∥ℓ2 :=
( ∞∑
n=1
|xn|2
)1/2
, ∀(xn) ∈ ℓ2.
It is known that it is a complete metric space; see Question 3 in Assignment 3. Let A be a
subset of ℓ2. Show that A is compact if and only if A is bounded, closed and for any ε > 0
there exists an integer N such that
∞∑
n=N+1
|xn|2 < ε, ∀(xn) ∈ A.
(Hint: Mimic the proof of Theorem 164 in lecture notes and Question 2 in Worksheet 8.)
Question 2. (4 + 5 = 9 points)
(i) Let B = {(a, b) : a, b ∈ R} and B′ := {(a, b) : a, b ∈ Q}. Show that B and B′ are
bases for some topologies on R. Do they generate the same topology?
(ii) Let C := {[a, b) : a, b ∈ R} and C ′ := {[a, b) : a, b ∈ Q}. Show that C and C ′ are
bases for some topologies on R. Do they generate the same topology?
Question 3. (3 + 4 + 4 + 4 = 15 points) Let X = [0, 1] and give X the cocountable
topology τc: a set U ⊂ X is open if and only if its complement is countable.
(i) Is τc Hausdorff?
(ii) Let (xn) be a sequence in X equipped with the cocountable topology. Show that
(xn) converges to x ∈ X if and only if xn = x for all sufficiently large n.
(iii) Let τd denote the discrete topology on X, i.e, every subset of X is open. Show that
τc and τd have the same convergent sequences, but τc ̸= τd. Hence knowing which
sequences converge does not allow one to determine the topology.
(iv) Let f : (X, τc)→ (X, τd) be given by f(x) = x. Is f continuous? Is f−1 : (X, τd)→
(X, τc) continuous?
Question 4. (3 + 3 + 3 + 3 = 12 points) For n ∈ N let Mn := {kn : k ∈ N} be the
set of positive multiples of n.
(a) Show that B := {Mn : n ∈ N} is a basis for a topology on N. This topology is
called the multiples topology on N.
(b) In the multiples topology, give six distinct neighborhoods of 10.
(c) In the multiples topology, does every k ∈ N have a smallest neighborhood? Explain.
(d) Prove or disprove: The multiples topology on N is Hausdorff.
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