MATH 645, HOMEWORK #3
DUE MONDAY, JUNE 5 AT 9AM.
Complete 5 of the following exercises for full credit. The ∗ exercises may be more challenging
than the others.
1. Prove that the circle S1 = {(x, y) ∈ R2 |x2 + y2 = 1} is homeomorphic to the square
T = {(x, y) | − 1 ≤ x ≤ 1, y = ±1} ∪ {(x, y) | − 1 ≤ y ≤ 1, x = ±1}.
2. Prove that the annulus A = {(x, y) ∈ R2 | 1 ≤ x2 + y2 ≤ 2} is homeomorphic to the cylinder
C = {(x, y, z) |x2 + y2 = 1 and 0 ≤ z ≤ 1}.
3. Prove that f : R` → R is continuous if and only if for every x0 ∈ R, we have limx→x+0 f(x) =
f(x0).
(Here, limx→x+0 f(x) is the limit as x approaches x0 from the right as defined in Calculus
classes. See the hint below.)
4. Let (Y,<) be a linearly ordered set with the order topology, and let X be any topological
space. Suppose that f, g : X → Y are continuous functions.
a. Show that C = {x ∈ X | f(x) ≤ g(x)} is a closed subset of X.
b. Prove that h : X → Y defined by h(x) = Max{f(x), g(x)} is continuous.
5. The text uses the notation Rω for the product
∏
n∈Z+ Xn with each Xn = R. Written in
another way
Rω = R× R× R× · · · .
The elements of Rω are sequences of real numbers, so x ∈ Rω has x = (xm)m∈Z+ = (x1, x2, x3, . . .).
For a sequence of elements xn ∈ Rω, n ≥ 1, write
xn = (xn1 , x
n
2 , x
n
3 , . . .).
Prove that the sequence (xn) ⊂ Rω, converges to the element x = (x1, x2, x3, . . .) ∈ Rω with
respect to the product topology on Rω if and only if
∀m ≥ 1, lim
n→+∞
xnm = xm.
6. Let R∞ ⊂ Rω be the subset consisting of sequences that are “eventually zero”, i.e. all
sequences (x1, x2, . . .) such that xi 6= 0 for only finitely many values of i. Determine the closure
of R∞ in Rω with respect to the
a. the product topology
b. the box topology.
7. Prove Theorem 19.4 from Munkres.
8.* Given a set X the countable complement topology, Tc, on X is the topology whose
open sets are those O ⊂ X such that X \ 0 is either all of X or is countable. (Here, as in Ch.
7 of Munkres, countable means countably infinite or finite.)
a. Prove that Tc is a topology on X. (You may use results from Ch. 7 of Munkres.)
b. Which sequences are convergent with respect to the countable complement topology?
(Give a necessary and sufficient condition.)
1
2 DUE MONDAY, JUNE 5 AT 9AM.
c. For R equipped with the countable complement topology, show that any x0 ∈ R is a
limit point of A = (0, 1), but there is no sequence in A \ {x0} that converges to x0 in
the countable complement topology.
Hints:
1. and 2. Visualize the homeomorphism, then write down a formula and carefully check continuity.
If your formula uses a piece-wise definition, then the Pasting Lemma (Theorem 18.3)
may be useful.
3. Recall the definition: limx→x+0 f(x) = L when ∀ > 0, ∃δ > 0 such that f(x) ∈ (f(x0)−
, f(x0) + ) whenever x ∈ (x0, x0 + δ).
It may be easiest to check continuity at individual points as in Theorem 18.1 (4).
4. Part a. may be useful for part b.
5. The definition of convergence for sequences in a topological space is on pg. 98 of Munkres.
6. Theorem 17.5 (b.) can be applied. Problem 5 may provide some insight into what the
closure would be for the product topology.