Math 403 Homework # 5
Due: Friday, 10-22-04
1. A non-empty set M⊆Rnis a subspace if Mis closed under addition and
scalar multiplication, i.e. for any ~x, ~y in Mand any r,s in Rwe have
r·~x +s·~y in Mas well.
Show that any subspace of Rnis a closed set.
2. Find the closure of the following sets:
(a) {~x ∈Rn:k~xk>1}
(b) {(x, y)∈R2:xy < 1}
(c) {(q, q) : q∈Q}
(d) {(x, sin x) : x > 0}
3. Let Aand Bbe compact subsets of Rn. Show that A∪Band A∩Bare
both compact.
4. Let A⊆Rnbe a compact set, and let h~akik≥1be a sequence of vectors in
Awhich does not have a limit. Show that this sequence has at least two
distinct cluster points.
∗5. (Extra credit) Let Aand Bbe two disjoint non-empty closed subsets of
Rn. Define the Hausdorff distance of Afrom B:
d(A, B) = inf {k~a −~
bk:~a ∈A, ~
b∈B}
(a) Show that if at least one of the two sets Aand Bis compact, then
d(A, B)>0.
(b) Give an example of two non-empty disjoint closed subsets Aand B
of R2with d(A, B) = 0.