Suggested problems - Analysis I
Mar 3
• Countable and uncountable sets
Mar 5
• Metric spaces: basic notions
Suggested problems: 2, 3, 4, 5, 8, 11 in Ch 2.
Additional problems
1. Show that Card R = Card (R \ N).
2. For each of the following sets, determine whether it is countable or uncountable.
(a) F := {A ⊂ R : Card A = Card N}
(b) G := {f : N → N : f is nonincreasing}
(c) H := {f : N → N : f is nondecreasing}
3. Define F : {0, 1}N → R by
F ((an )n∈N ) =
∞
X
2an
n=1
for (an )n∈N ∈ {0, 1}N .
3n
Show that F is injective, and conclude that R is uncountable.
4. Let X = {0, 1}N . For x = (xn )n∈N and y = (yn )n∈N in X, define




0,
if x = y,
d(x, y) =



2−n , where n = min{k ∈ N : x 6= y }.
k
Show that d is a metric on X. In addition, show that
E := {(an )n∈N : a1 = 0}
is open and closed.
1
k
2
5. Let X be a metric space. A point p is said to be
(a) an exterior point of E if there exists a neighborhood Nr (p) such that E ∩ Nr (p) = ∅;
(b) a boundary point of E if every neighborhood Nr (p) intersects both E and E c .
Show that
(a) if x is an exterior point of E if and only if x is an interior point of E c ;
(b) for E ⊂ X, every point x is exactly one of the following:
(1) an interior point of E
(2) an exterior point of E
(3) a boundary point of E
6. Let X be a metric space. For E ⊂ X, denote the set of all boundary points of E by ∂E.
Show that
(a) E is open if and only if E ∩ ∂E = ∅;
(b) E is closed if and only if ∂E ⊂ E;
(c) E is open if and only if E c is closed.