Homework 5
Math 501
Due October 3, 2014
Exercise 1
Let M = Q. Show that the set S = Q ∩ [0, 1] is closed and bounded but not
compact. [Hint: Consider the set A = {x ∈ S : x2 < 1/2}. Argue that this set
is bounded above, construct a sequence that converges to its supremum, and go
from there.]
Exercise 2
Let a < b be real numbers. Prove that [a, b) is not compact. Deduce that
there cannot exist a homeomorphism between [a, b] and [a, b) .
Exercise 3
The set of all bounded, real-valued sequences is often denoted
`∞ = {(xn ) ⊂ R : sup |xn | < ∞}.
n≥1
It is not hard to show that the map d : `∞ × `∞ → R
d(x, y) = sup |xn − yn |
n≥1
gives a metric on `∞ , so we may regard `∞ as a metric space. In this exercise,
we show that the closed unit ball B 1 (0) = {x ∈ `∞ : d(x, 0) ≤ 1} in `∞ is not
compact. This, like Exercise 1, is in stark contrast to the situation in Rn , where
by the Heine-Borel theorem, the closed unit ball is in fact compact.
Define for each n ≥ 1 the sequence
en = (0, . . . , 0, 1, 0, . . .)
which has 1 in coordinate n and 0 in all other coordinates. It is obvious that
each en ∈ `∞ lies in the closed unit ball B 1 (0) of `∞ . Show that the sequence
1
(en ) has no convergent subsequence. [Hint: First prove that if a sequence in a
metric space converges, then it is Cauchy. What is d(en , em ) for any n 6= m?]
Exercise 4
Let f : M → N be a homeomorphism and let A ⊂ M . Prove that
(a) f (int A) = int f (A),
(b) f (A) = f (A),
(c) f (∂A) = ∂f (A).
Exercise 5
Consider the following subsets of R2 :
A = {x ∈ R2 : 1 ≤ |x| ≤ 2},
D = {x ∈ R2 : |x| ≤ 1},
S = {x = (x1 , x2 ) ∈ R2 : |x2 | ≤ 1},
H = {x = (x1 , x2 ) ∈ R2 : x2 ≥ 0}.
Prove that
(a) A and S are not homeomorphic,
(b) A and D are not homeomorphic,
(c) S and H are not homeomorphic.
[In fact no pair of these sets is homeomorphic.]
Exercise 6
Let A, B, C be sets with A ⊂ B and let f : A → C and g : B → C be
functions. We say that f extends to g (or that g extends f ) if f (a) = g(a)
for all a ∈ A.
Let M be a metric space, let S ⊂ M be a subset, and let f : S → R be a
uniformly continuous function. Let p ∈ S.
(a) Suppose that (pn ) ⊂ S is a sequence that converges to p. Prove that f (pn )
converges in R. [Hint: prove that the sequence f (pn ) is Cauchy.]
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(b) Suppose that (qn ) ⊂ S is some other sequence that converges to p. Prove
that f (qn ) converges to the same point to which f (pn ) from part (a) converges.
(c) Define a function f : S → R by f (x) = f (x) for x ∈ S and for p ∈ S \ S
define
f (p) = lim f (pn )
n→∞
where (pn ) ⊂ S is any sequence that converges to p ∈ S. Parts (a) and
(b) above show that f is well-defined; that is, f (p) does not depend on the
choice of sequence. Prove that f is uniformly continuous.
(d) Prove that f is the unique continuous function defined on S that extends
f.
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