Homework
M472
Fall 2012
Exercise 1. Let X = R with the euclidean topology, and Q ⊆ R. Describe the
topological space R/Q, obtained by identifying all points of Q together. (Note
this is NOT the group quotient!!)
Exercise 2 (Cone construction). Given any topological space X, we define the
cone over X:
X × [0, 1]
,
CX :=
∼
where [0, 1] is considered with the euclidean topology, and ∼ is generated by
(x1 , 1) ∼ (x2 , 1), for all x1 , x2 ∈ X.
1. Prove that the cone over a circle is homeomorphic to a closed disk.
2. In general let S n = {x s.t.||x|| = 1 ⊆ Rn+1 } be the n dimensional sphere.
Prove that
n+1
,
CS n ∼
=D
n+1
where D
= {x s.t.||x|| ≤ 1 ⊆ Rn+1 } denotes the n + 1 dimensional
closed disk.
n
n
Exercise 3. Consider D and S n−1 = ∂D = {x s.t.||x|| = 1 ⊆ Rn }. Prove
that:
n
D ∼ n
n =S .
∂D
Exercise 4. Prove that P2 is homeomorphic to the one-point compactification
of an open Moebius strip, or equivalently, to the identification space obtained
from a closed Mobius strip by collapsing its boundary to a point.