PMATH 333 Assignment 7
due Wednesday, November 3rd by 10pm (Waterloo local time)
In order to receive full credits, you must show all the work leading to your solution. You may of
course work together with your classmates, but you must write up the solutions independently.
Problems
#1 Let E ⊆ Rd be a non-empty compact set. The following questions are independent.
(a) (4 pts) Prove that for all ε > 0 there exist N ∈ N and x1, · · · , xN ∈ E such that
E ⊆ ∪Nj=1Bε(xj).
(b) (4 pts) Suppose {Fi}i∈I is an arbitrary collection of closed subsets of Rd such that
Fi ⊆ E for all i ∈ I and that Fi1 ∩ · · · ∩ FiN 6= ∅ for any N ∈ N and i1, · · · , iN ∈ I.
Prove that ∩i∈IFi 6= ∅.
#2 Prove that the following sets are compact.
(a) (5 pts) E = {0} ∪
(
∪∞n=1B 1
n
(an)
)
⊆ Rd, where (an)N is a given sequence in Rd con-
verging to 0.
(b) (7 pts) E = {0} ∪ { 1
n
| n ∈ N} ∪ { 1
n
+ 1
m
| n,m ∈ N} ⊆ R.
#3 (10 pts) Let E,K be disjoint subsets of Rd, with E closed and K compact. Prove that
there exists α > 0 such that ‖x− y‖ ≥ α for all x ∈ E and y ∈ K. (First prove that for all
y ∈ K there exists ry > 0 such that Bry(y) ∩ E = ∅.)
#4 (8 pts) Let E be an uncountable subset of Rd. Prove that there exists x ∈ E such that
Bδ(x) ∩ E is uncountable for all δ > 0.
#5 (12 pts) Let E be a subset of Rd. Prove that the following two statements are equivalent.
(i) E is compact.
(ii) For any infinite subset F ⊆ E, there exists x ∈ E such that Bδ(x) ∩ F is infinite for
all δ > 0.
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